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Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (auto i:s) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) { + char c, r = 0, n = 0; + x = 0; + for (;;) { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 800 + +#define TVAL int +#define TLAZY int +#define TLIM 4100 + +TVAL ZERO_VAL = 0; +TLAZY ZERO_LAZY = 0; + +struct SegTree +{ + void UpdateValForUpdateOrLazy(TVAL& a, TLAZY v) + { + a += v; + } + + void UpdateLazyForUpdateOrLazy(TLAZY& a, TLAZY v) + { + a += v; + } + + TVAL CombVals(TVAL v1, TVAL v2) + { + return(max(v1, v2)); + } + + int N, sz; + TVAL tree[TLIM]; + TLAZY lazy[TLIM]; + + SegTree() {} + + SegTree(int _N) + { + Init(_N); + } + + void Init(int _N) + { + N = _N; + for (sz = 1; sz < N; sz <<= 1); + Clear(); + } + + void Clear() + { + int i; + Fox(i, sz << 1) + tree[i] = ZERO_VAL; + Fox(i, sz << 1) + lazy[i] = ZERO_LAZY; + } + + void Prop(int i) + { + TLAZY v = lazy[i]; + lazy[i] = ZERO_LAZY; + UpdateValForUpdateOrLazy(tree[i], v); + if (i < sz) + { + int c1 = i << 1, c2 = c1 + 1; + UpdateLazyForUpdateOrLazy(lazy[c1], v); + UpdateLazyForUpdateOrLazy(lazy[c2], v); + } + } + + void Comp(int i) + { + int c1 = i << 1, c2 = c1 + 1; + tree[i] = CombVals(tree[c1], tree[c2]); + } + + TVAL Query( + int a, int b, + int i = 1, int r1 = 0, int r2 = -1 + ) { + if (r2 < 0) + { + Max(a, 0); + Min(b, sz - 1); + if (a > b) + return ZERO_VAL; + r2 = sz - 1; + } + Prop(i); + if (a <= r1 && r2 <= b) + return(tree[i]); + int m = (r1 + r2) >> 1, c = i << 1; + TVAL ret = ZERO_VAL; + if (a <= m) + ret = CombVals(ret, Query(a, b, c, r1, m)); + if (b > m) + ret = CombVals(ret, Query(a, b, c + 1, m + 1, r2)); + return(ret); + } + + void Update( + int a, int b, + TLAZY v, + int i = 1, int r1 = 0, int r2 = -1 + ) { + if (r2 < 0) + { + Max(a, 0); + Min(b, sz - 1); + if (a > b) + return; + r2 = sz - 1; + } + Prop(i); + if (a <= r1 && r2 <= b) + { + UpdateLazyForUpdateOrLazy(lazy[i], v); + Prop(i); + return; + } + int m = (r1 + r2) >> 1, c = i << 1; + if (a <= m) + Update(a, b, v, c, r1, m); + if (b > m) + Update(a, b, v, c + 1, m + 1, r2); + Prop(c), Prop(c + 1), Comp(i); + } +}; + +struct Event +{ + int x, y1, y2, c; + bool s; +}; + +const bool operator<(const Event& a, const Event& b) +{ + return(mp(a.x, !a.s) < mp(b.x, !b.s)); +} + +int S, N; +int C[LIM], P[LIM][4]; + +bool IsBetween(int a, int b, int p, bool ex) +{ + if (b < a) + b += INF; + if (p < a) + p += INF; + return ex ? a < p && p < b : a <= p && p <= b; +} + +int GetPosAfter(int a, int p) +{ + if (p < a) + p += INF; + return(p - a); +} + +int SolveForLine(vector h) +{ + int i, j, s; + // compare all pieces against dividing line, and assemble line sweep events + int base = 0; + vector CY; + vector E; + Fox(i, N) + { + // full intersection? + if ( + (IsBetween(P[i][0], P[i][2], h[0], 0) || IsBetween(P[i][3], P[i][1], h[0], 0)) && + (IsBetween(P[i][0], P[i][2], h[1], 0) || IsBetween(P[i][3], P[i][1], h[1], 0)) + ) + { + base += C[i]; + continue; + } + // look for orientation of line segments such that at least one spans crosses from the 1st to the 2nd half + Fox(s, 2) + { + int p[4]; + memcpy(p, P[i], sizeof(p)); + if (s) + reverse(p, p + 4); + // check which points are on their required halves + bool bx[2], by[2]; + Fox(j, 2) + { + bx[j] = IsBetween(h[0], h[1], p[j * 2], 0); + by[j] = IsBetween(h[1], h[0], p[j * 2 + 1], 0); + } + // neither line segment is entirely valid? + if ((!bx[0] || !by[0]) && (!bx[1] || !by[1])) + continue; + assert(bx[0] + by[0] + bx[1] + by[1] >= 3); // other one must be at least half-valid + // map points to positions on their halves + int x[2], y[2]; + Fox(j, 2) + { + x[j] = bx[j] ? GetPosAfter(h[0], p[j * 2]) : 2 * INF * (j ? 1 : -1); + y[j] = by[j] ? -GetPosAfter(h[1], p[j * 2 + 1]) : 2 * INF * (j ? 1 : -1); + } + assert(x[0] < x[1] && y[0] < y[1]); + E.pb({ x[0], y[0], y[1], C[i], 1 }); + E.pb({ x[1], y[0], y[1], C[i], 0 }); + CY.pb(y[0]), CY.pb(y[1]); + break; + } + } + // compress Y-coordinates + sort(All(CY)); + int K = unique(All(CY)) - CY.begin(); + CY.resize(K); + // line sweep + SegTree ST(K); + sort(All(E)); + Foxen(e, E) + { + e.y1 = lower_bound(All(CY), e.y1) - CY.begin(); + e.y2 = lower_bound(All(CY), e.y2) - CY.begin(); + // left edge of a rectangle? + if (e.s) + { + ST.Update(e.y1, K - 1, e.c); + continue; + } + // right edge of a rectangle + ST.Update(e.y2, e.y2, ST.Query(e.y2, K - 1) - ST.Query(e.y2, e.y2)); + ST.Update(e.y2 + 1, K - 1, -e.c); + } + return(base + ST.Query(0, K - 1)); +} + +int ProcessCase() +{ + int i, j; + // input + Read(S), Read(N); + Fox(i, N) + { + Read(C[i]); + Fox(j, 4) + { + int x, y; + Read(x), Read(y); + if (!x) + P[i][j] = y; + else if (y == S) + P[i][j] = S + x; + else if (x == S) + P[i][j] = 3 * S - y; + else + P[i][j] = 4 * S - x; + } + // normalize lines + if (IsBetween(P[i][1], P[i][0], P[i][2], 1)) + swap(P[i][0], P[i][1]); + if (IsBetween(P[i][2], P[i][3], P[i][0], 1)) + swap(P[i][2], P[i][3]); + } + // consider all possible dividing lines + int ans = 0; + Fox(i, N) + { + Fox(j, 2) + Max(ans, SolveForLine({ P[i][j * 2], P[i][j * 2 + 1] })); + } + return(ans); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/finals/cake_cutting_committee.in b/2020/finals/cake_cutting_committee.in new file mode 100644 index 0000000000000000000000000000000000000000..bdbde10c118643a57098f6ce63eece1a629ff370 --- /dev/null +++ b/2020/finals/cake_cutting_committee.in @@ -0,0 +1,17085 @@ +112 +10 3 +20 0 4 10 4 0 6 10 6 +30 0 8 10 8 0 9 10 9 +40 2 0 2 10 5 0 5 10 +4 3 +10 0 1 1 0 0 2 2 0 +20 0 2 2 4 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cake for a very special occasion. She's starting off with a square cake, though she'd like it cut down into a more unique shape. When viewed from above, it can be represented as a square on a 2D plane, with opposite corners at coordinates \((0, 0)\) and \((S, S)\). + +Out of countless contenders, Melody has interviewed \(N\) certified cake-cutting consultants, asking how the cake should be cut down. The \(i\)th consultant, who has a competence value of \(C_i\), has proposed that two cuts should be performed: the first being a line segment between coordinates \((X_{i,1}, Y_{i,1})\) and \((X_{i,2}, Y_{i,2})\), and the second between \((X_{i,3}, Y_{i,3})\) and \((X_{i,4}, Y_{i,4})\). Each of these four points is exactly on a side of the cake (but not on a corner), the two points describing each line segment are not on the same side of the cake, and the two line segments do not share any points in common (in other words, they don't intersect nor have any equal endpoints). Therefore, the two cuts would split the cake into three non-empty polygonal sections. Of those, the middle section (the one between the two cuts) would become their **proposed cake shape**, with the other two sections discarded. + +For example, the following diagram illustrates a cake with \(S = 10\) and a consultant's pair of proposed cuts (from \((4, 10)\) to \((2, 0)\) and from \((0, 9)\) to \((2, 10)\)), with their resulting proposed cake shape highlighted in blue: + +{{PHOTO_ID:186542789854750}} + +Melody wants to hire one or more of the consultants to form a committee, whose members will then deliberate on the final shape of the cake. That is, she calls for a completely competent consulting committee capable of coming to a consensus on a commendable cake-cutting configuration. Clearly. + +She wants the sum of the hired consultants' competence values to be as large as possible, no matter how many of them she has to hire. However, she won't allow a pair of consultants to both be part of the committee if their cake-cutting proposals *fundamentally differ*, which is the case if their proposed cake shapes do not share any points in common (in other words, if those two polygons are entirely disjoint, including at their edges and vertices). + +For example, the following diagram additionally includes another consultant's pair of proposed cuts (from (7, 10) to (10, 7) and from (10, 5) to (5, 10)), with their proposed cake shape highlighted in red. Since these two proposed cake shapes do not overlap at all, the two proposals fundamentally differ, meaning that if one of these consultants is hired, the other cannot also be hired: + +{{PHOTO_ID:3047534892140325}} + +On the other hand, each of the following cakes features a pair of proposals which do not fundamentally differ, due to the proposed cake shapes sharing at least one point in common (meaning that, for each cake, the two consultants could be hired together): + +{{PHOTO_ID:139722534273576}} + +Help Melody determine the maximum combined competence of consultants who can form the committee, such that no two of them have fundamentally differing cake-cutting proposals. + +# Constraints + +\(1 \le T \le 80\) +\(1 \le N \le 800\) +\(2 \le S \le 100,000,000\) +\(1 \le C_i \le 500,000\) +\(0 \le X_{i,j}, Y_{i,j} \le S\) + +The sum of \(N\) across all special occasions is at most 20,000. + +# Input + +Input begins with an integer \(T\), the number of special occasions. For each occasion, there are \(N + 1\) lines. The first line contains 2 space-separated integers, \(S\) and \(N\). \(N\) lines follow, the \(i\)th of which contains the 9 space-separated integers \(C_i\), \(X_{i,1}\), \(Y_{i,1}\), \(X_{i,2}\), \(Y_{i,2}\), \(X_{i,3}\), \(Y_{i,3}\), and \(X_{i,4}\), \(Y_{i,4}\). + + +# Output + +For the \(i\)th special occasion, print a line containing *"Case #i: "*, followed by a single integer, the maximum combined competence of cake-cutting consultants who can form the committee. + + +# Explanation of Sample + +The first cake (along with the consultants' proposed cake shapes) looks as follows: + +{{PHOTO_ID:2722961511304050}} + +The 1st and 2nd consultants can't both be hired together, but either can be hired alongside the 3rd consultant. So, the best Melody can do is to hire the 2nd and 3rd consultants for a combined competence of \(30 + 40 = 70\). + +The second cake looks as follows: + +{{PHOTO_ID:408936830244710}} + +Melody can hire all three consultants. + +The third cake looks as follows: + +{{PHOTO_ID:386528679257402}} + +The fourth cake looks as follows: + +{{PHOTO_ID:863939674419710}} + +The fifth cake looks as follows: + +{{PHOTO_ID:424906348725741}} diff --git a/2020/finals/cake_cutting_committee.out b/2020/finals/cake_cutting_committee.out new file mode 100644 index 0000000000000000000000000000000000000000..abbcedf23e8e452985fa225b8d4b78846d0249c5 --- /dev/null +++ b/2020/finals/cake_cutting_committee.out @@ -0,0 +1,112 @@ +Case #1: 70 +Case #2: 60 +Case #3: 60 +Case #4: 14 +Case #5: 11 +Case #6: 15 +Case #7: 26 +Case #8: 62 +Case #9: 59 +Case #10: 33 +Case #11: 10 +Case #12: 4 +Case #13: 8160 +Case #14: 7207 +Case #15: 7 +Case #16: 59 +Case #17: 41008 +Case #18: 52938 +Case #19: 17272 +Case #20: 250000000 +Case #21: 81000 +Case #22: 81000 +Case #23: 195153474 +Case #24: 157757408 +Case #25: 148427301 +Case #26: 33268941 +Case #27: 31942779 +Case #28: 29651063 +Case #29: 31340125 +Case #30: 29857414 +Case #31: 29451361 +Case #32: 29524220 +Case #33: 27477273 +Case #34: 30685666 +Case #35: 30626812 +Case #36: 27372492 +Case #37: 29689094 +Case #38: 5215 +Case #39: 55269 +Case #40: 3394 +Case #41: 1870 +Case #42: 3902 +Case #43: 1656 +Case #44: 22432 +Case #45: 61590 +Case #46: 2693 +Case #47: 11236 +Case #48: 4765 +Case #49: 11836 +Case #50: 25810 +Case #51: 17702 +Case #52: 69830 +Case #53: 13183 +Case #54: 39633 +Case #55: 16641 +Case #56: 20295 +Case #57: 8134 +Case #58: 371 +Case #59: 4361 +Case #60: 15012 +Case #61: 667 +Case #62: 7749 +Case #63: 7587 +Case #64: 66822 +Case #65: 13309 +Case #66: 8544 +Case #67: 62661 +Case #68: 22366 +Case #69: 1235 +Case #70: 12477 +Case #71: 12164 +Case #72: 7031 +Case #73: 21198 +Case #74: 15176 +Case #75: 6046 +Case #76: 1149 +Case #77: 35941 +Case #78: 810 +Case #79: 23259 +Case #80: 1335 +Case #81: 2950 +Case #82: 7170 +Case #83: 41925 +Case #84: 8782 +Case #85: 1107 +Case #86: 15565 +Case #87: 17094 +Case #88: 1250 +Case #89: 27818 +Case #90: 10246 +Case #91: 5690 +Case #92: 17200 +Case #93: 6868 +Case #94: 7621 +Case #95: 24280 +Case #96: 15790 +Case #97: 806 +Case #98: 24283 +Case #99: 21818 +Case #100: 636 +Case #101: 39706 +Case #102: 9505 +Case #103: 16829 +Case #104: 59786 +Case #105: 5017 +Case #106: 23559 +Case #107: 48806 +Case #108: 7429 +Case #109: 35980 +Case #110: 62511 +Case #111: 2200 +Case #112: 45468 diff --git a/2020/finals/cake_cutting_committee_sol.md b/2020/finals/cake_cutting_committee_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..f00f83246ab59d78d014071629cdc8706e9229c7 --- /dev/null +++ b/2020/finals/cake_cutting_committee_sol.md @@ -0,0 +1,18 @@ +We can number the possible positions of line segment endpoints from \(1\) to \(4(N-1)\) in clockwise order around the edge of the square, and then simply map all of the given line segment endpoints to these indices, noting that only their relative order is relevant for determining which proposed cake shapes overlap with one another. + +It's possible to show that, for any valid set of proposed cake shapes, there must exist at least one *key line segment* amongst them such that every polygon in the set overlaps with that line segment. We'll therefore consider all \(2N\) possible key line segments. + +For a given key line segment, each polygon either doesn't overlap with it at all (and so should be ignored), overlaps with all of it (and so may always be used), or overlaps with only part of it. For polygons falling into this final category, we're interested in the maximum combined value of any set of them such that each pair of them overlaps. + +The key line segment splits the full set of possible endpoint indices into two sections, which we'll call the X and Y sections. We can map each polygon's endpoints to new indices along these two sections, both numbered starting from the same ("first") endpoint of the key line segment (meaning that one section's sequence of indices is equivalent to increasing original indices, while the other section's is equivalent to decreasing original indices). If a polygon has 3 endpoints in one section and 1 endpoint in the other (as opposed to 2 in each), then we'll shift one of them over to one of the key line segment's endpoints without changing which other polygons it overlaps with. + +The result is that, for each line segment of each polygon, one endpoint will have been mapped to a new index in the X section, with the other mapped to a new index in the Y section. This allows us to represent each polygon by a pair of intervals \([x_1, x_2]\) and \([y_1, y_2]\) (consisting of its two X section indices and its two Y section indices, respectively). With this representation, we can then observe that a pair of polygons don't overlap if and only one of the polygons' X and Y intervals are both strictly to the right (or strictly to the left) of the other polygon's respective X and Y intervals. + +If we now visualize each such polygon as a rectangle on a 2D plane, with coordinates based on its X and Y intervals (in particular, with lower-left corner \((x_1, y_1)\) and upper-right corner \((x_1, y_1)\)), we're interested in the maximum combined value of any set of them such that no rectangle is strictly both above and to the right of any other rectangle. This may be accomplished in \(O(N log(N))\) time with a line sweep over the x-coordinates. We'll let \(M_y\) be the maximum combined value of a valid set of rectangles encountered so far, such that the lowest of their top-right corners encountered so far has y-coordinate `y`, and maintain this with the help of a segment tree: + +- When we encounter the left edge of a rectangle with bottom edge `y_1`, we should increase each of \(M_{y_1..\infty}\) by that rectangle's value (as this rectangle may be added to any such set). +- When we encounter the right edge of a rectangle with top edge `y_2`, we should first set \(M_{y_2}\) to \(\max(M_{(y_2+1)..\infty})\) (as including this rectangle in any such set would decrease its lowest top-right corner to \(y_2\)). We should then decrease each of \(M_{(y_2+1)..\infty}\) by that rectangle's value (as this rectangle may no longer be added to any such set). + +Repeating the above process for all \(2N\) possible key line segments yields a solution with a time complexity of \(O(N^2 log(N))\) (though a less efficient, \(O(N^3)\) implementation would also suffice). + +For further reading, if we consider mapping the line segment endpoint indices to positions around the circumference of a circle instead (essentially treating the original square as a circle), each proposed cake shape is known as a *circle trapezoid*. The intersection graph of these polygons (a graph in which there's an edge between two pieces if and only if they overlap) is then known as a [*circle trapezoid graph*](https://en.wikipedia.org/wiki/Trapezoid_graph#Circle_trapezoid_graphs). The problem of computing the maximum clique weight in such a graph (equivalent to what's asked for in Cake-Cutting Committee), and other related problems, are explored further in [this paper](https://www.sciencedirect.com/science/article/pii/S0166218X96000133?via%3Dihub). diff --git a/2020/finals/cryptoconference.cpp b/2020/finals/cryptoconference.cpp new file mode 100644 index 0000000000000000000000000000000000000000..0c6da747f7243e7b29684ee75d5777f070d57959 --- /dev/null +++ b/2020/finals/cryptoconference.cpp @@ -0,0 +1,158 @@ +// Cryptoconference +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) { + char c, r = 0, n = 0; + x = 0; + for (;;) { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define MOD 1000000007 + +int N, K; +LL cur, ans; +set S; + +// returns the # of valid intervals with start positions in: +// (this interval's start, next interval's start] +LL Count(set::iterator I) +{ + // get key points from this interval and the next one + int a = I->x + 1; + I++; + int b = I->x; + int c = I->y - 1; + // count intervals which start in [a, b] and end no later than c + int v1 = c - b; + int v2 = c - a; + return (LL)(v1 + v2) * (v2 - v1 + 1) / 2; +} + +// inserts interval [a, b] into the set, while updating the total # of valid intervals +void Insert(int a, int b) +{ + set::iterator I, J; + I = S.lower_bound(mp(a, -1)); + // is the new interval obsolete (covers existing interval)? + if (a <= I->x && I->y <= b) + return; + // erase any existing obsolete intervals (covering the new one) + if (I->x > a) + I--; + for (;;) + { + if (I->y < b) + break; + J = I, I--; + cur -= Count(J) + Count(I); + S.erase(J); + cur += Count(I); + } + // insert the new interval + cur -= Count(I); + S.insert(mp(a, b)); + cur += Count(I); + I++; + cur += Count(I); +} + +LL ProcessCase() +{ + int i; + // input + Read(N), Read(K); + // init + S.clear(); + S.insert(mp(-1, -1)); + S.insert(mp(K, K + 1)); + ans = 1; + cur = Count(S.begin()); + // process intervals + Fox(i, N) + { + int s, d; + Read(s), Read(d); + Insert(s, s + d); + ans = ans * (cur % MOD) % MOD; + } + return(ans); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %lld\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/finals/cryptoconference.in b/2020/finals/cryptoconference.in new file mode 100644 index 0000000000000000000000000000000000000000..00f85880f11c5cc57836a8d70b43e19bf4cde3cc --- /dev/null +++ b/2020/finals/cryptoconference.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a50294e575c083ccf35a9f1298d80e950fb199d926cc2203e9d7f80970603d6e +size 14176322 diff --git a/2020/finals/cryptoconference.md b/2020/finals/cryptoconference.md new file mode 100644 index 0000000000000000000000000000000000000000..0ae7bf455b1b1d3ce78b4434f06ae34c7f1cc781 --- /dev/null +++ b/2020/finals/cryptoconference.md @@ -0,0 +1,41 @@ +Crys just invented a new cryptocurrency and is planning to present it at \(T\) blockchain conferences around the world. The conference schedules are already on their way to being finalized, but Crys is considering how she might get her own talks in at the last minute. + +A certain conference is \(K\) hours long, with talks to start and end on the hour. Each talk will span a non-empty inclusive time interval of hours \([x, y]\), such that \(x\) and \(y\) are integers and \(0 \le x \lt y \le K\). \(N\) talks are gradually being added to the schedule, one after another, with the \(i\)th added talk scheduled to span the time interval \([S_i, S_i + D_i]\). The \(N\) talks are not necessarily added in order of starting or ending time. Their time intervals may arbitrarily overlap with one another, and are not necessarily distinct. + +Each time a talk is added to the schedule, Crys will then consider submitting a request for her own talk to also be included. This involves proposing a time interval for her talk, which must similarly be an integral interval \([x, y]\) such that \(0 \le x \lt y \le K\). Crys suspects that her request would be rejected if the inclusion of her talk might prevent an attendee from hearing another talk due to her proposed time interval entirely subsuming an already-scheduled one. As such, after the first \(i\) talks have been added to the schedule, she will *not* propose a time interval \([x, y]\) if there exists any existing talk \(j\) \((1 \le j \le i)\) such that \(x \le S_j\) and \(S_j + D_j \le y\). + +Letting \(C_i\) be the number of different time intervals which Crys could validly propose for her talk after the first \(i\) talks have been added to the schedule, please determine the value of \((C_1 * C_2 * ... * C_N)\) modulo 1,000,000,007. + +# Constraints + +\(1 \le T \le 95\) +\(1 \le N \le 300,000\) +\(1 \le K \le 1,000,000,000\) +\(0 \le S_i < K\) +\(1 \le D_i \le K\) +\(1 \le S_i + D_i \le K\) + +The sum of \(N\) across all conferences is at most 1,500,000. + +# Input + +Input begins with an integer \(T\), the number of conferences. For each conference there are \(N + 1\) lines. The first line contains the space-separated integers \(N\) and \(K\). \(N\) lines follow, the \(i\)th of which contains the space-separated integers \(S_i\) and \(D_i\). + + +# Output + +For the \(i\)th conference, print a line containing *"Case #i: "*, followed by a single integer, the product of \(C_{1..N}\) modulo 1,000,000,007. + + + + + +# Explanation of Sample + +In the first conference, once a talk spanning time interval \([1, 2]\) exists, Crys can only propose time intervals \([0, 1]\) and \([2, 3]\) for her talk without subsuming \([1, 2]\). Therefore, \(C = [2]\). + +In the second conference, after the first talk (spanning time interval \([1, 3]\)) has been added to the schedule, Crys can propose any of the time intervals \([0, 1]\), \([0, 2]\), \([1, 2]\), or \([2, 3]\). After the introduction of the second talk (spanning time interval \([0, 2]\)), she can only propose \([0, 1]\), \([1, 2]\), or \([2, 3]\). Therefore, \(C = [4, 3]\), and the final answer is (4 * 3) modulo 1,000,000,007 = 12. + +In the third conference, \(C = [8, 4]\). + +In the fourth conference, \(C = [4126, 3631, 3239, 2687, 2687]\). diff --git a/2020/finals/cryptoconference.out b/2020/finals/cryptoconference.out new file mode 100644 index 0000000000000000000000000000000000000000..babd5d979dd6293af14712f57f4c6bbc5fdec70d --- /dev/null +++ b/2020/finals/cryptoconference.out @@ -0,0 +1,110 @@ +Case #1: 2 +Case #2: 12 +Case #3: 32 +Case #4: 352996146 +Case #5: 751550872 +Case #6: 20 +Case #7: 83444858 +Case #8: 572887886 +Case #9: 598916604 +Case #10: 470820377 +Case #11: 174927279 +Case #12: 896919730 +Case #13: 541761571 +Case #14: 515822691 +Case #15: 228409848 +Case #16: 659858375 +Case #17: 190419874 +Case #18: 948002069 +Case #19: 495164089 +Case #20: 989289949 +Case #21: 313062055 +Case #22: 317384968 +Case #23: 381355941 +Case #24: 861250172 +Case #25: 21268787 +Case #26: 426473611 +Case #27: 174566443 +Case #28: 270050563 +Case #29: 247721801 +Case #30: 128678835 +Case #31: 141554145 +Case #32: 589771144 +Case #33: 703459805 +Case #34: 770102403 +Case #35: 32994412 +Case #36: 98689617 +Case #37: 146465775 +Case #38: 429027202 +Case #39: 249562463 +Case #40: 939240223 +Case #41: 366445856 +Case #42: 629076290 +Case #43: 856574351 +Case #44: 816923530 +Case #45: 988573584 +Case #46: 227744671 +Case #47: 136671501 +Case #48: 552031554 +Case #49: 637297699 +Case #50: 0 +Case #51: 407393317 +Case #52: 648160821 +Case #53: 453730949 +Case #54: 626850629 +Case #55: 77514078 +Case #56: 776033559 +Case #57: 981403229 +Case #58: 164846153 +Case #59: 486727330 +Case #60: 181718458 +Case #61: 794033109 +Case #62: 741574342 +Case #63: 145879918 +Case #64: 171838401 +Case #65: 417158876 +Case #66: 725642265 +Case #67: 472767853 +Case #68: 740493091 +Case #69: 89444180 +Case #70: 608421866 +Case #71: 393203295 +Case #72: 879337785 +Case #73: 915216546 +Case #74: 290037518 +Case #75: 248932214 +Case #76: 953715588 +Case #77: 167356836 +Case #78: 317540564 +Case #79: 991575711 +Case #80: 879687486 +Case #81: 322773218 +Case #82: 589228118 +Case #83: 466166842 +Case #84: 748094367 +Case #85: 248333831 +Case #86: 515789417 +Case #87: 951341566 +Case #88: 948722510 +Case #89: 517082433 +Case #90: 535310061 +Case #91: 102636817 +Case #92: 805901472 +Case #93: 546788335 +Case #94: 684359382 +Case #95: 454605467 +Case #96: 310565738 +Case #97: 700636353 +Case #98: 266780658 +Case #99: 383812708 +Case #100: 664630757 +Case #101: 289228410 +Case #102: 957511402 +Case #103: 584046200 +Case #104: 58349857 +Case #105: 730036244 +Case #106: 945387917 +Case #107: 744001995 +Case #108: 854694249 +Case #109: 644309315 +Case #110: 801361727 diff --git a/2020/finals/cryptoconference_sol.md b/2020/finals/cryptoconference_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..5cb6661cd3343532ca00e35efab0c939766ba27f --- /dev/null +++ b/2020/finals/cryptoconference_sol.md @@ -0,0 +1,9 @@ +We'll maintain a set \(R\) of all "relevant" talk time intervals. An interval \([x, y]\) is considered irrelevant if ignoring it does not change the set of time intervals which may be proposed, which is the case if and only if there exists another interval which is inclusively contained within it (an interval \([x', y']\) such that \(x \le x'\) and \(y' \le y\)). + +We'll maintain \(R\) ordered by intervals' start times, in strictly increasing order. Note that this also means that the intervals' end times must be strictly increasing (otherwise, they cannot all be relevant). For convenience, we'll also begin by including dummy intervals \([-1, -1]\) and \([K, K+1]\) in \(R\), whose significance is explained below. From there, updating the set for each new interval may be done relatively straightforwardly in amortized \(O(log(N))\) time, with the new interval either already being irrelevant, or being relevant and causing 0 or more existing intervals to become irrelevant. + +What remains is maintaining the corresponding answer for \(R\) — that is, the number of time intervals which may be proposed without subsuming any interval in \(R\). For this purpose, let \(F([x_1, y_1], [x_2, y_2])\) be the number of different intervals \([x_p, y_p]\) which may be proposed such that \(x_1 < x_p \le x_2\), given that \([x_1, y_1]\) and \([x_2, y_2]\) form a pair of consecutive relevant intervals in \(R\). Note that \(F([x_1, y_1], [x_2, y_2])\) does not depend on any intervals in \(R\) besides those two, and may be easily computed in \(O(1)\) time using the fact that the only constraints are \(x_1 < x_p \le x_2\) and \(x_p < y_p < y_2\). + +We can observe that the sum of \(F([x_1, y_1], [x_2, y_2])\) over all pairs of consecutive relevant intervals (of which there are \(|R| - 1\)) is equal to the total answer — for example, this is the case when \(R\) consists only of the initial pair of dummy intervals \([-1, -1]\) and \([K, K+1]\), with additional intervals simply partitioning the set of possible start times. We can then maintain this sum with no additional time complexity as we go. For example, when inserting an interval \(X\) between two existing intervals \(A\) and \(B\), we should increase the sum by \(F(A, X) + F(X, B) - F(A, B)\), with deletions handled similarly. + +This gives us an algorithm with a time complexity of \(O(N log(N))\). diff --git a/2020/finals/pond_precipitation.cpp b/2020/finals/pond_precipitation.cpp new file mode 100644 index 0000000000000000000000000000000000000000..d17056048fee6dd47580018c256081ea51490fd7 --- /dev/null +++ b/2020/finals/pond_precipitation.cpp @@ -0,0 +1,178 @@ +// Pond Precipitation +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 32 +#define LIM2 500 +#define MOD 1000000007 + +int N; +int D[LIM], C[LIM]; +int ch[LIM2][LIM2]; +int dyn[LIM][LIM2]; + +int Exp(int a, int b) +{ + LL p = a, v = 1; + while (b) + { + if (b & 1) + v = v * p % MOD; + p = p * p % MOD; + b >>= 1; + } + return(v); +} + +void Add(int& a, int b) +{ + a = (a + b) % MOD; +} + +int Prod(int a, int b) +{ + return((LL)a * b % MOD); +} + +int Div(int a, int b) +{ + return Prod(a, Exp(b, MOD - 2)); +} + +int ProcessCase() +{ + int i, j, k; + // input + Read(N); + Fox(i, N) + Read(D[i]); + // compute max prefix capacities + int p = 0, s = 0; + Fox(i, N + 1) + { + if (i == N || D[i] < D[p]) + { + FoxI(j, p, i - 1) + s += D[j] - D[p]; + FoxI(j, p + (p ? 1 : 0), i) + C[j] = s; + p = i; + } + } + // DP + Fill(dyn, 0); + dyn[0][0] = 1; + Fox(i, N) + { + Fox(j, (i ? C[i - 1] : 0) + 1) + { + Fox(k, C[i] - j + 1) + Add(dyn[i + 1][j + k], Prod(dyn[i][j], ch[j + k][k])); + } + } + // compute answer + int ans = 0; + Fox(i, s + 1) + Add(ans, Div(dyn[N][i], Exp(N, i))); + return(ans); +} + +int main() +{ + int T, t; + int i, j; + // precompute choose table + ch[0][0] = 1; + Fox1(i, LIM2 - 1) + { + Fox(j, i + 1) + { + ch[i][j] = ch[i - 1][j]; + if (j) + Add(ch[i][j], ch[i - 1][j - 1]); + } + } + // testcase loop + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/finals/pond_precipitation.in b/2020/finals/pond_precipitation.in new file mode 100644 index 0000000000000000000000000000000000000000..9b84aaf123deb55f4a96d86c15ba1195a69ce25d --- /dev/null +++ b/2020/finals/pond_precipitation.in @@ -0,0 +1,129 @@ +64 +2 +2 1 +2 +1 2 +1 +30 +3 +3 1 2 +3 +1 3 2 +5 +3 5 4 1 2 +8 +3 7 5 8 2 6 1 4 +15 +11 29 12 10 13 28 15 6 5 25 17 8 9 3 24 +2 +30 1 +2 +1 30 +16 +10 28 18 17 20 19 21 29 5 14 11 8 7 25 3 27 +19 +22 27 26 14 24 17 10 9 30 12 19 28 1 13 25 23 8 20 18 +20 +30 15 29 14 28 13 27 12 26 11 25 10 24 9 23 8 22 7 21 6 +30 +1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 +30 +3 30 22 20 14 17 27 19 24 1 23 25 5 12 11 15 29 13 2 8 21 16 26 28 18 4 6 7 9 10 +30 +30 23 9 4 14 10 20 29 26 19 3 22 16 13 24 11 2 15 18 8 7 17 5 28 27 12 6 1 21 25 +29 +23 24 25 30 14 26 22 12 27 19 18 7 8 28 13 9 15 29 10 4 20 17 6 11 3 21 5 1 2 +29 +21 26 6 14 19 27 9 8 23 3 20 4 25 12 17 16 10 22 13 30 24 11 18 15 7 5 1 29 2 +28 +21 28 18 26 10 30 13 19 6 17 24 23 11 3 2 29 4 27 22 14 9 8 1 12 7 20 15 16 +28 +4 12 28 9 15 5 1 17 26 20 27 22 13 29 3 2 21 25 14 6 24 23 10 18 30 16 8 11 +27 +30 24 25 27 20 29 26 15 11 28 23 13 16 10 14 19 3 6 17 8 22 18 2 12 5 1 4 +27 +27 22 21 1 5 7 26 23 13 18 14 28 25 19 29 6 11 3 17 30 24 10 12 16 4 15 8 +26 +22 11 2 16 1 25 15 9 29 12 4 21 8 26 24 18 6 30 13 19 5 23 17 20 7 3 +26 +2 25 29 20 21 27 28 8 23 10 18 17 26 22 24 13 7 12 3 5 14 4 1 15 6 9 +6 +18 25 26 8 10 20 +13 +16 26 25 2 30 27 19 24 22 9 6 4 14 +14 +1 7 20 26 17 11 30 14 15 27 28 5 24 12 +4 +30 7 26 2 +11 +30 24 29 9 18 17 25 13 8 11 4 +16 +10 28 18 17 20 19 21 29 5 14 11 8 7 25 3 27 +5 +30 21 18 9 8 +13 +19 13 24 7 8 21 16 20 3 15 2 30 11 +2 +1 25 +8 +13 12 23 22 25 20 30 15 +13 +20 8 13 3 17 9 30 22 28 2 21 5 12 +19 +22 27 26 14 24 17 10 9 30 12 19 28 1 13 25 23 8 20 18 +9 +30 27 28 13 5 6 2 19 15 +4 +12 25 23 11 +11 +24 6 29 16 14 22 11 1 7 15 2 +20 +11 29 12 10 13 28 15 2 3 25 21 17 8 9 5 24 27 16 23 26 +17 +11 7 6 9 20 14 21 30 23 19 24 8 27 25 2 15 12 +3 +24 20 2 +18 +30 16 25 8 6 23 14 27 29 3 12 18 22 19 9 15 13 11 +6 +17 15 14 12 3 11 +12 +4 22 23 15 13 1 19 8 12 27 14 21 +7 +1 16 25 13 23 7 6 +1 +3 +19 +7 27 24 30 23 29 8 28 25 19 13 21 12 18 1 11 16 10 4 +7 +10 29 30 9 8 5 11 +6 +29 27 25 17 7 2 +6 +29 25 12 10 15 9 +7 +26 21 24 19 20 14 13 +17 +7 25 27 17 21 28 29 24 10 2 4 19 5 8 26 3 11 +16 +18 28 14 7 26 21 17 10 16 13 12 4 8 2 6 1 +16 +23 15 30 16 4 29 1 9 14 2 12 11 22 24 21 17 +6 +30 18 24 17 8 5 +5 +27 30 17 15 5 +15 +29 26 20 28 10 17 19 9 15 5 6 4 8 11 3 +7 +26 27 24 10 17 2 4 +13 +26 22 30 18 28 14 9 20 3 10 5 2 4 +8 +15 20 24 26 14 5 1 2 +1 +26 +4 +17 5 22 7 +6 +28 24 17 5 1 3 diff --git a/2020/finals/pond_precipitation.md b/2020/finals/pond_precipitation.md new file mode 100644 index 0000000000000000000000000000000000000000..d7f356e80a990cf9910938e131e6d9aad19d5cd5 --- /dev/null +++ b/2020/finals/pond_precipitation.md @@ -0,0 +1,76 @@ +Cherry the turtle lives in a long, narrow pond. When cross-sectionally viewed from the side, the floor of the pond can be divided into \(N\) columns, each flat and 1 metre wide. The \(i\)th column from the left is at a depth of \(D_i\) metres below ground level, and all \(N\) column depths are distinct. The pond is surrounded by dirt (at ground level), just to the left of column 1 and the right of column \(N\). Cherry loves to sit on the floor in the leftmost column and bask as much as possible — but only when it's dry. Fortunately for her, the pond is initially devoid of water. + +For example, if \(D = [3, 5, 4, 1, 2]\), the pond would look as follows (with Cherry's position indicated in green): + +{{PHOTO_ID:311416436611683}} + +Unusually enormous drops of rainwater are about to begin falling into the pond, one by one, each onto a random column (drawn uniformly at random from the set of \(N\) columns). Each drop contains 1 cross-sectional square metre of water, and obeys the following process: + + 1. It falls until it hits a flat *surface*. This surface consists of the highest (least deep) point in its column (which might be either the floor of the pond itself or an existing layer of water atop it), as well as all contiguous equally-deep points to the left and right of it. + 2. If that surface is immediately surrounded by any deeper columns to its left and/or right, then the drop's water flows down onto one of them. If there's only one such adjacent deeper column, then the drop flows onto that one. If the columns to the left and right are both deeper than the surface (note that this can only occur if the surface consists of a single column), then the drop's water all flows to the *left* one. Either way, once the drop flows into a different column, the process repeats back from Step 1. + 3. Otherwise, the drop's 1 cross-sectional square metre of water spreads itself out evenly across the entire surface it landed on (which will never cause that surface to "overflow" by becoming strictly higher than the columns to its left or right). + +Cherry will be forced to stop basking as soon as a non-zero amount of water comes to rest atop column 1. She doesn't mind if raindrops fall directly on column 1 but then immediately flow away from it. + +For example, if a raindrop were to fall in column 4 of the above pond, it would flow to the left (onto column 3), and then flow left again to settle in column 2: + +{{PHOTO_ID:3533638723394837}} + +If a raindrop then fell in column 1, it would flow to the right and spread itself out over the surface spanning columns 2 and 3: + +{{PHOTO_ID:299743861289784}} + +If a raindrop then fell in column 5, it would remain there: + +{{PHOTO_ID:1094784447639382}} + +Another raindrop falling in column 5 (or *any* column) would then yield the following state: + +{{PHOTO_ID:747829032476688}} + +And one more raindrop falling in any column would yield the following state (forcing Cherry to stop basking due to column 1 becoming submerged in water): + +{{PHOTO_ID:477696676534167}} + +Determine the expected number of drops which will fall before Cherry is forced to stop basking. This will always occur after a finite number of drops have fallen, and before any water overflows the pond. + +Let this expected number of drops be represented as a quotient of integers \(p/q\) in lowest terms. Output the value of this quotient modulo 1,000,000,007 — in other words, output the unique integer \(x\) such that \(0 \le x \lt 1,000,000,007\) and \(p \equiv x*q\text{ }(\text{modulo }1,000,000,007)\). + + +# Constraints + +\(1 \le T \le 50\) +\(1 \le N \le 30\) +\(1 \le D_i \le 30\) +\(D_{1..N}\) are distinct + +The sum of \(N\) across all ponds is at most 1000. + + +# Input + +Input begins with an integer \(T\), the number of ponds. For each pond, there are 2 lines. The first line contains a single integer \(N\). The second line contains the \(N\) space-separated integers \(D_{1..N}\). + + +# Output + +For the \(i\)th pond, print a line containing *"Case #i: "*, followed by a single integer, the expected number of drops which will fall before the leftmost column becomes covered in water, expressed as a quotient of integers modulo 1,000,000,007. + + +# Explanation of Sample + +The first pond initially looks as follows: + +{{PHOTO_ID:1173260263089522}} + +If the first raindrop falls in column 1, it will settle there, and if it falls in column 2 instead, it will flow into column 1. Either way, Cherry's position will be submerged and she'll be forced to stop basking. + +The second pond initially looks as follows: + +{{PHOTO_ID:127321889008596}} + +The first raindrop will always settle in column 2, leaving Cherry uninterrupted. However, the second raindrop will then always land on the surface spanning both columns, filling both up to a depth of 0.5 metres below ground level and thus forcing Cherry to stop basking. + +The third pond only has a single column, and Cherry will need to stop basking as soon as the first raindrop falls in it. + +The fourth pond may end up in various states depending on which columns the raindrops fall in. If the first drop falls in either column 1 or 2, then it will immediately settle in column 1, while if it falls in column 3, it will remain there and it will take a second drop to force Cherry to stop basking. The answer is therefore \(\frac{2}{3} 1 + \frac{1}{3} 2 = \frac{4}{3}\). \(4 \equiv 333,333,337 * 3\text{ }(\text{modulo }1,000,000,007)\), meaning that \(333,333,337\) should be outputted. diff --git a/2020/finals/pond_precipitation.out b/2020/finals/pond_precipitation.out new file mode 100644 index 0000000000000000000000000000000000000000..67be11d4d38ce4a1e96f10bccb580780182281a9 --- /dev/null +++ b/2020/finals/pond_precipitation.out @@ -0,0 +1,64 @@ +Case #1: 1 +Case #2: 2 +Case #3: 1 +Case #4: 333333337 +Case #5: 4 +Case #6: 772800010 +Case #7: 600238862 +Case #8: 777291572 +Case #9: 1 +Case #10: 30 +Case #11: 937030791 +Case #12: 413793850 +Case #13: 839435693 +Case #14: 436 +Case #15: 443547255 +Case #16: 396227790 +Case #17: 466103174 +Case #18: 817567450 +Case #19: 617077188 +Case #20: 328627652 +Case #21: 923842801 +Case #22: 461892853 +Case #23: 120289842 +Case #24: 898611294 +Case #25: 639276210 +Case #26: 153140977 +Case #27: 224 +Case #28: 64313891 +Case #29: 492753435 +Case #30: 937030791 +Case #31: 1 +Case #32: 147204623 +Case #33: 25 +Case #34: 748059186 +Case #35: 716331815 +Case #36: 413793850 +Case #37: 351458863 +Case #38: 25 +Case #39: 510316599 +Case #40: 840006893 +Case #41: 943180527 +Case #42: 1 +Case #43: 249621861 +Case #44: 746767720 +Case #45: 981645522 +Case #46: 85 +Case #47: 1 +Case #48: 509775157 +Case #49: 3252335 +Case #50: 1 +Case #51: 646090541 +Case #52: 822157443 +Case #53: 357912021 +Case #54: 532071843 +Case #55: 723030647 +Case #56: 799725660 +Case #57: 4 +Case #58: 124032548 +Case #59: 677744521 +Case #60: 752783508 +Case #61: 893366069 +Case #62: 1 +Case #63: 64313891 +Case #64: 194444447 diff --git a/2020/finals/pond_precipitation_sol.md b/2020/finals/pond_precipitation_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..cc30f22ba94e21dab4791b78f8091f82d59a44e3 --- /dev/null +++ b/2020/finals/pond_precipitation_sol.md @@ -0,0 +1,5 @@ +Let \(C_i\) be the maximum number of raindrops which can fall in the first \(i\) columns without causing any water to settle atop column 1. We can compute \(C_{1..N}\) in \(O(N)\) time by iterating over the columns from left to right, looking for each new highest (least deep) column, and considering filling the columns before it (back to the previous highest column) with water up to the depth of that previous highest column. + +Then, let \(\textit{DP}_{i,d}\) be the number of different ways for \(d\) raindrops to fall in the first \(i\) columns without causing any water to settle atop column 1. \(\textit{DP}_{0,0}\) = 1, with the remaining values computable using dynamic programming as follows. We can consider all triples \((i, d, d')\) in increasing order of \(i\) such that \(d\) drops have fallen in the first \(i\) columns (such that \(d \le C_i\)) and \(d'\) drops will fall in column \(i + 1\) (such that \(d + d' \le C_{i+1}\)). This allows us to increase \(\textit{DP}_{i+1,d+d'}\) by \(\textit{DP}_{i,d} * {d+d' \choose d'}\), given that the new \(d'\) drops may be mixed in anywhere amongst the total \(d+d'\) drops. Overall, this process takes \(O(N * \sum(D_{1..N})^2)\) time. + +Finally, let \(S_d\) be the probability that, once \(d\) drops have fallen, no water has yet settled atop column 1. Note that \(S_d = \textit{DP}_{N,d} / N^i\). The answer may then be computed as \(\sum(S_{0..C_N})\). diff --git a/2020/finals/somebody_elses_problem.cpp b/2020/finals/somebody_elses_problem.cpp new file mode 100644 index 0000000000000000000000000000000000000000..60faf03f35b4782d93f85ba7b25155c95b9679bd --- /dev/null +++ b/2020/finals/somebody_elses_problem.cpp @@ -0,0 +1,152 @@ +// Somebody Else's Problem +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (auto i:s) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) { + char c, r = 0, n = 0; + x = 0; + for (;;) { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define MOD 1000000007 +#define LIM 1000001 + +int N, ans; +vector ch[LIM]; +int dyn1[LIM]; // max. len. of downward i -> leaf +int dyn2[LIM]; // max. len. of downward root -> leaf outside i's subtree +int dyn3[LIM]; // max. disjoint len. of downward root -> leaf plus other path ongoing from i's parent + +void rec1(int i) +{ + // recurse, and compute dyn1[i] + dyn1[i] = 0; + Foxen(c, ch[i]) + { + rec1(c); + Max(dyn1[i], dyn1[c] + 1); + } +} + +void rec2(int i, int d) +{ + // compute answer for i + ans = ans * (LL)max(dyn1[i] + dyn2[i] + (!i ? 0 : 1), dyn3[i]) % MOD; + // compute max 2 (dyn1[c] + 1, c) pairs + PR m1, m2; + m1 = m2 = mp(0, -1); + Foxen(c, ch[i]) + { + PR p = mp(dyn1[c] + 1, c); + if (p > m1) + m2 = m1, m1 = p; + else if (p > m2) + m2 = p; + } + // compute dyn2[c] and dyn3[c], and recurse + Foxen(c, ch[i]) + { + int m = (m1.y == c ? m2 : m1).x; + dyn2[c] = max(d + m, dyn2[i]); + dyn3[c] = max(dyn3[i] + 1, dyn2[i] + m + 1 + (!i ? 0 : 1)); + rec2(c, d + 1); + } +} + +int ProcessCase() +{ + int i, j; + // input + Read(N); + Fox1(i, N - 1) + { + Read(j), j--; + ch[j].pb(i); + } + // DP + ans = 1; + rec1(0); + rec2(0, 0); + // reset + Fox(i, N) + ch[i].clear(); + return(ans); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/finals/somebody_elses_problem.in b/2020/finals/somebody_elses_problem.in new file mode 100644 index 0000000000000000000000000000000000000000..fcbe8c16ef659e319885dbe30e45809c1c7fb57b --- /dev/null +++ b/2020/finals/somebody_elses_problem.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:42fb026894d3a2c67d12bff5c8c7605976525e17fb94ed2dfc18ae0b5a91a494 +size 29992845 diff --git a/2020/finals/somebody_elses_problem.md b/2020/finals/somebody_elses_problem.md new file mode 100644 index 0000000000000000000000000000000000000000..8737678bd37c9c255b60e29c52d073d0e32aaaec --- /dev/null +++ b/2020/finals/somebody_elses_problem.md @@ -0,0 +1,44 @@ +H Combinator (HC) is a business incubator that provides seed funding to startups. It operates on the core value that "nothing is somebody else's problem" (since businesses slow down when tasks get perpetually passed from person to person). HC currently supports \(T\) companies. As a managing director hoping to weed out some bad investments, you would like to assess how well the structure of each company holds up to this important principle. + +Suppose a company has \(N\) employees, numbered \(1\) to \(N\) in decreasing order of seniority. Employee \(1\) is the CEO, while each other employee \(i\) \((i \ge 2)\) has a manager, employee \(M_i\) \((M_i \lt i)\). + +When employee \(i\) is assigned a task to complete, they may instead reassign it to one of the following employees: + + 1. One of their direct reports (any employee \(j\) such that \(M_j = i\)), if any exist + 2. Their manager (employee \(M_i\)), if \(i \ge 2\) + 3. The CEO (employee \(1\)), if \(i \ge 2\) + +That employee may then reassign the task to yet another employee, and so on. However, an employee may never be reassigned a task that has ever been previously assigned to them, as the lack of responsibility would become too apparent. For example, an employee and manager cannot repeatedly pass a task back and forth. + +Let \(R_i\) be the maximum number of times that a task could be reassigned from one employee to another if it were initially assigned to employee \(i\). For each company, you would like to calculate the product \((R_1 * R_2 * ... * R_N)\), modulo 1,000,000,007. + +# Constraints + +\(1 \le T \le 100\) +\(2 \le N \le 1,000,000\) +\(1 \le M_i \lt i\) + +The sum of \(N\) across all companies is at most 6,000,000. + + +# Input + +Input begins with an integer \(T\), the number of companies. For each company there are 2 lines. The first line contains the single integer \(N\). The second line contains \(N - 1\) space-separated integers, \(M_{2..N}\). + + +# Output + +For the \(i\)th company, print a line containing *"Case #i: "*, followed by a single integer, the product of \(R_{1..N}\) modulo 1,000,000,007. + + +# Explanation of Sample + +For the first company, if the CEO is initially assigned a task, all they can do is reassign it to one of their reports, who cannot then assign it back. On the other hand, if another employee is initially assigned a task, they can reassign it to the CEO, who can then assign it to the other remaining employee. Therefore, \(R = [1, 2, 2]\), and the final answer is (1 * 2 * 2) modulo 1,000,000,007 = 4. + +For the second company, no matter which employee is initially assigned a task, it can then be reassigned twice. For example, employee 2 can assign their task to their report (employee 3), who can then assign it to the CEO. Therefore, \(R = [2, 2, 2]\). + +For the third company, \(R = [2, 2, 3, 3]\). + +For the fourth company, \(R = [3, 6, 6, 6, 7, 7, 7, 7, 8, 8]\). + +For the fifth company, \(R = [4, 6, 6, 7, 8, 6, 7, 7, 8, 7, 7, 8, 8, 9, 9]\). diff --git a/2020/finals/somebody_elses_problem.out b/2020/finals/somebody_elses_problem.out new file mode 100644 index 0000000000000000000000000000000000000000..16a203ad6cf66d2791ea499b924c74805cbf25ac --- /dev/null +++ b/2020/finals/somebody_elses_problem.out @@ -0,0 +1,109 @@ +Case #1: 4 +Case #2: 8 +Case #3: 36 +Case #4: 99574272 +Case #5: 801542729 +Case #6: 512 +Case #7: 110638080 +Case #8: 100755024 +Case #9: 609548256 +Case #10: 449377614 +Case #11: 697200545 +Case #12: 40681884 +Case #13: 801335875 +Case #14: 912420599 +Case #15: 828211555 +Case #16: 222945837 +Case #17: 794621467 +Case #18: 539189349 +Case #19: 153951545 +Case #20: 403356442 +Case #21: 158834216 +Case #22: 757493143 +Case #23: 124049818 +Case #24: 157599195 +Case #25: 210009530 +Case #26: 934651967 +Case #27: 846283835 +Case #28: 135977128 +Case #29: 96550718 +Case #30: 460241648 +Case #31: 168595112 +Case #32: 212089364 +Case #33: 533764564 +Case #34: 887365886 +Case #35: 959297767 +Case #36: 518707324 +Case #37: 926754190 +Case #38: 800571094 +Case #39: 352762395 +Case #40: 446732699 +Case #41: 137078595 +Case #42: 82460488 +Case #43: 63988313 +Case #44: 368079615 +Case #45: 874039636 +Case #46: 259792328 +Case #47: 680495316 +Case #48: 310927912 +Case #49: 159825100 +Case #50: 757078864 +Case #51: 11192700 +Case #52: 815983871 +Case #53: 221642594 +Case #54: 881701383 +Case #55: 233697937 +Case #56: 842482923 +Case #57: 765406592 +Case #58: 989866696 +Case #59: 898464379 +Case #60: 345934457 +Case #61: 451190753 +Case #62: 322496398 +Case #63: 475160557 +Case #64: 996799426 +Case #65: 211994997 +Case #66: 690657310 +Case #67: 971619993 +Case #68: 440397393 +Case #69: 481512798 +Case #70: 186452849 +Case #71: 708281413 +Case #72: 357202043 +Case #73: 777137428 +Case #74: 841177015 +Case #75: 825117615 +Case #76: 853698291 +Case #77: 594949860 +Case #78: 404255967 +Case #79: 458058760 +Case #80: 589833391 +Case #81: 311188915 +Case #82: 858709805 +Case #83: 457598660 +Case #84: 291250389 +Case #85: 89919554 +Case #86: 964724850 +Case #87: 87366696 +Case #88: 253283539 +Case #89: 417747428 +Case #90: 10644788 +Case #91: 95418217 +Case #92: 560671530 +Case #93: 326047040 +Case #94: 463949173 +Case #95: 905018983 +Case #96: 201781788 +Case #97: 955583192 +Case #98: 319780485 +Case #99: 514572911 +Case #100: 92757427 +Case #101: 829713327 +Case #102: 367443664 +Case #103: 932977208 +Case #104: 247500328 +Case #105: 387055634 +Case #106: 191909169 +Case #107: 421840147 +Case #108: 730451318 +Case #109: 493473473 diff --git a/2020/finals/somebody_elses_problem_sol.md b/2020/finals/somebody_elses_problem_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..eed573fd9e1b62d3887f97d86f0fcff3cb580956 --- /dev/null +++ b/2020/finals/somebody_elses_problem_sol.md @@ -0,0 +1,18 @@ +The employees can be considered nodes of a tree rooted at node 1 (the CEO), with each node \(i\)'s set of children \(C_i\) consisting of all nodes \(j\) such that \(M_j = i\). We'll additionally let \(D_i\) be the depth of node \(i\), and \(S_i\) be its set of siblings. + +We can observe that \(R_i\) corresponds to either of the following: + +- The maximum length of any single simple path starting at node \(i\) (potentially going upwards to some ancestor and then back downwards) +- 1 plus the maximum combined length of any node-disjoint pair of simple paths, the first one starting at node \(i\) (potentially going upwards to some ancestor and then back downwards) and the second starting at the root (and only going downwards) + +We'll approach computing these lengths and sums with dynamic programming. + +First, let \(\textit{DP1}_i\) be the maximum length of any path starting at node \(i\) and contained within \(i\)'s subtree. \(\textit{DP1}_{1..N}\) may be computed in standard fashion based on the recurrence \(\textit{DP1}_i = \max(c \in C_i | \textit{DP1}_c + 1)\). + +Next, let \(\textit{DP2}_i\) be the maximum length of any path starting at the root and never entering node \(i\)'s subtree. If \(i\) is the root, then \(\textit{DP2}_i = 0\). Otherwise, \(\textit{DP2}_i = \max(D_{M_i} + \max(s \in S_i | \textit{DP1}_s + 1), \textit{DP2}_{M_i})\). + +Now, let \(\textit{DP3}_i\) be 1 plus the maximum combined length of any node-disjoint pair of simple paths, the first one starting at node \(i\) but not including any of \(i\)'s children, and the second starting at the root. If \(i\) is the root, then \(\textit{DP3}_i = 0\). Otherwise, \(\textit{DP3}_i = \max(\textit{DP3}_{M_i} + 1, \textit{DP2}_{M_i} + \max(s \in S_i | \textit{DP1}_s + 1) + 2)\). + +Finally, if \(i\) is the root, then \(R_i = \textit{DP1}_i\). Otherwise, \(R_i = \max(\textit{DP1}_i + \textit{DP2}_i + 1, \textit{DP3}_i)\). + +We can compute all of the above values in \(O(N)\) time through two recursive (or iterative) passes over the tree. diff --git a/2020/finals/spider_spring.cpp b/2020/finals/spider_spring.cpp new file mode 100644 index 0000000000000000000000000000000000000000..a5496d29bbe8148acd63ee2bd3d9035377be2aad --- /dev/null +++ b/2020/finals/spider_spring.cpp @@ -0,0 +1,310 @@ +// Spider Spring +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) { + char c, r = 0, n = 0; + x = 0; + for (;;) { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else if (r) + break; + } + if (n) + x = -x; + return(1); +} + +void ReadSeq(int* V, int N, int K) +{ + int i, A, B, C, D; + Fox(i, K) + Read(V[i]); + Read(A), Read(B), Read(C), Read(D); + FoxI(i, K, N - 1) + V[i] = ((LL)A * V[i - 2] + (LL)B * V[i - 1] + C) % D + 1; +} + +#define MOD 1000000007 +#define LIM 1000001 + +#define TVAL LL +#define TLAZY pair // (set, increase) +#define TLIM 2100000 + +TVAL ZERO_VAL = 0; +TLAZY ZERO_LAZY = mp(0, 0); + +struct SegTree +{ + void UpdateValForUpdateOrLazy(TVAL& a, TLAZY v) + { + if (v.x) + a = v.x; + if (v.y) + a += v.y; + } + + void UpdateLazyForUpdateOrLazy(TLAZY& a, TLAZY v) + { + if (v.x) + a.x = v.x; + if (v.y) + a.y += v.y; + } + + TVAL CombVals(TVAL v1, TVAL v2) + { + return(v1 + v2); + } + + int N, sz; + TVAL tree[TLIM]; + TLAZY lazy[TLIM]; + + SegTree() {} + + SegTree(int _N) + { + Init(_N); + } + + void Init(int _N) + { + N = _N; + for (sz = 1; sz < N; sz <<= 1); + Clear(); + } + + void Clear() + { + int i; + Fox(i, sz << 1) + tree[i] = ZERO_VAL; + Fox(i, sz << 1) + lazy[i] = ZERO_LAZY; + } + + void Prop(int i) + { + TLAZY v = lazy[i]; + lazy[i] = ZERO_LAZY; + UpdateValForUpdateOrLazy(tree[i], v); + if (i < sz) + { + int c1 = i << 1, c2 = c1 + 1; + UpdateLazyForUpdateOrLazy(lazy[c1], v); + UpdateLazyForUpdateOrLazy(lazy[c2], v); + } + } + + void Comp(int i) + { + int c1 = i << 1, c2 = c1 + 1; + tree[i] = CombVals(tree[c1], tree[c2]); + } + + TVAL Query( + int a, int b, + int i = 1, int r1 = 0, int r2 = -1 + ) { + if (r2 < 0) + r2 = sz - 1; + Prop(i); + if (a <= r1 && r2 <= b) + return(tree[i]); + int m = (r1 + r2) >> 1, c = i << 1; + TVAL ret = ZERO_VAL; + if (a <= m) + ret = CombVals(ret, Query(a, b, c, r1, m)); + if (b > m) + ret = CombVals(ret, Query(a, b, c + 1, m + 1, r2)); + return(ret); + } + + void Update( + int a, int b, + TLAZY v, + int i = 1, int r1 = 0, int r2 = -1 + ) { + if (r2 < 0) + r2 = sz - 1; + Prop(i); + if (a <= r1 && r2 <= b) + { + UpdateLazyForUpdateOrLazy(lazy[i], v); + Prop(i); + return; + } + int m = (r1 + r2) >> 1, c = i << 1; + if (a <= m) + Update(a, b, v, c, r1, m); + if (b > m) + Update(a, b, v, c + 1, m + 1, r2); + Prop(c), Prop(c + 1), Comp(i); + } + + void UpdateOne(int i, TLAZY v) { + i += sz; + UpdateValForUpdateOrLazy(tree[i], v); + while (i > 1) + { + i >>= 1; + Comp(i); + } + } +}; + +int N, M, K; +int H[LIM], X[LIM], Y[LIM], Z[LIM], W[LIM]; +set LS; + +// 0: height of column i +// 1: sum of A endpoint weights with endpoint value i +// 2: sum of A endpoint weighted values with endpoint value i +// 3: sum of B endpoint weights with endpoint value i +// 4: sum of B endpoint weighted values with endpoint value i +SegTree ST[5]; + +void UpdateHeights(int a, int b, int h) +{ + ST[0].Update(a, b, mp(h, 0)); +} + +void UpdateLineSegment(int i, int d) // d = 1 (add) / -1 (remove) +{ + if (i < 0 || i + 1 >= N) + return; + int a = ST[0].Query(i, i), b = ST[0].Query(i + 1, i + 1); + if (a > b) + swap(a, b); + LL w = (LL)(i + 1) * (N - i - 1); + ST[1].UpdateOne(a, mp(0, w % MOD * d)); + ST[2].UpdateOne(a, mp(0, a * w % MOD * d)); + ST[3].UpdateOne(b, mp(0, w % MOD * d)); + ST[4].UpdateOne(b, mp(0, b * w % MOD * d)); + if (d > 0) + LS.insert(i); + else + LS.erase(i); +} + +int ProcessCase() +{ + int i; + // input + Read(N), Read(M), Read(K); + ReadSeq(H, N, K); + ReadSeq(X, M, K); + ReadSeq(Y, M, K); + ReadSeq(Z, M, K); + ReadSeq(W, M, K); + // init segment trees, line segment set, and horizontal sum + Fox(i, 5) + ST[i].Init(!i ? N : LIM); + LS.clear(); + LL horiz = 0; + Fox(i, N) + { + UpdateHeights(i, i, H[i]); + UpdateLineSegment(i - 1, 1); + horiz = (horiz + (LL)i * (N - i)) % MOD; + } + // process events + int ans = 1; + Fox(i, M) + { + // update + int x = X[i] - 1; + int y = min(x + Y[i], N) - 1; + int z = Z[i]; + auto I = LS.lower_bound(x - 1); + while (I != LS.end() && *I <= y) + { + int k = *I; + I++; + UpdateLineSegment(k, -1); + } + UpdateHeights(x, y, z); + UpdateLineSegment(x - 1, 1); + UpdateLineSegment(y, 1); + // query + int w = W[i]; + LL d1 = ST[4].Query(w + 1, LIM - 1) - ST[2].Query(w + 1, LIM - 1); + LL d2 = ST[3].Query(w + 1, LIM - 1) - ST[1].Query(w + 1, LIM - 1); + LL vert = ((d1 - d2 % MOD * w) % MOD + MOD) % MOD; + ans = (LL)ans * (horiz + vert) * 2 % MOD; + } + return(ans); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/finals/spider_spring.in 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The spring features \(N\) columns of rocks in a row, numbered \(1\) to \(N\) from left to right. Each rocky column \(i\) has a width of 1 cm and a height of \(H_i\) cm. + +The spring always has some global water level. The rocky columns can each be below the water, at water level, or protrude above its surface. A spider in a given column must float on the surface of the water (if the rock is below water level) or stand atop the rock (if it's at or above water level). This means that, when the water level is W cm, the **effective elevation** at column \(i\) is \(max(H_i, W)\) cm. + +For example, the following diagram illustrates a spring with \(H = [2, 1, 3, 1]\) and \(W = 2\), with the effective elevations outlined in black: + +{{PHOTO_ID:1583104035206937}} + +Water spiders sometimes want to travel from one column \(a\) to another column \(b\). This involves starting horizontally in the middle of column \(a\), at its effective elevation, and walking along the outline of the effective elevations until ending up horizontally in the middle of column \(b\). This may include walking left/right across the tops of rocks or the surface of the water, as well as up/down along the sides of the protruding rocks. Note that a spider's route will never take it left of column \(1\), right of column \(N\), below the surface of the water, nor into mid-air. + +For example, the path which a water spider would follow from column 1 to column 4 (having a length of 5 cm) is indicated in red below: + +{{PHOTO_ID:862340924576901}} + +Unfortunately for the spring's inhabitants, frequent earthquakes occur, altering the heights of the rocky columns and the water. The spiders would like to survey the accessibility of the spring after each earthquake. + +Specifically, \(M\) earthquakes are about to occur, one after another. As part of the \(i\)th earthquake, the height of each rocky column \(c\) such that \(X_i \le c \le min(X_i + Y_i - 1, N)\) will become equal to \(Z_i\) cm, and the global water level will become \(W_i\) cm. After each earthquake \(i\), the spiders are interested in the current sum of travel distances (in cm) over all \(N*(N - 1)\) ordered pairs of starting and ending columns — we'll call this quantity \(S_i\). + +Please determine the value of \((S_1 * S_2 * ... * S_M)\) modulo 1,000,000,007. + +In order to reduce the size of the input, the above values will not all be provided explicitly. Instead, you'll be given the first \(K\) values \(H_{1..K}\), \(X_{1..K}\), \(Y_{1..K}\), \(Z_{1..K}\), and \(W_{1..K}\), as well as the five quadruples of constants \((A_H, B_H, C_H, D_H)\), \((A_X, B_X, C_X, D_X)\), \((A_Y, B_Y, C_Y, D_Y)\), \((A_Z, B_Z, C_Z, D_Z)\), and \((A_W, B_W, C_W, D_W)\), and must then compute \(H_{(K+1)..N}\), \(X_{(K+1)..M}\), \(Y_{(K+1)..M}\), \(Z_{(K+1)..M}\), and \(W_{(K+1)..M}\) as follows: + + \(H_i = ((A_H * H_{i-2} + B_H * H_{i-1} + C_H) \text{ modulo } D_H) + 1\) for \(i > K\) + \(X_i = ((A_X * X_{i-2} + B_X * X_{i-1} + C_X) \text{ modulo } D_X) + 1\) for \(i > K\) + \(Y_i = ((A_Y * Y_{i-2} + B_Y * Y_{i-1} + C_Y) \text{ modulo } D_Y) + 1\) for \(i > K\) + \(Z_i = ((A_Z * Z_{i-2} + B_Z * Z_{i-1} + C_Z) \text{ modulo } D_Z) + 1\) for \(i > K\) + \(W_i = ((A_W * W_{i-2} + B_W * W_{i-1} + C_W) \text{ modulo } D_W) + 1\) for \(i > K\) + + +# Constraints + +\(1 \le T \le 90\) +\(2 \le N \le 1,000,000\) +\(2 \le M \le 1,000,000\) +\(2 \le K \le min(N, M)\) +\(0 \le A_H, B_H, C_H, A_X, B_X, C_X, A_Y, B_Y, C_Y, A_Z, B_Z, C_Z, A_W, B_W, C_W \le 1,000,000\) +\(1 \le D_H, D_Z, D_W \le 1,000,000\) +\(1 \le D_X, D_Y \le N\) +\(1 \le H_i \le D_H\) +\(1 \le X_i \le D_X\) +\(1 \le Y_i \le D_Y\) +\(1 \le Z_i \le D_Z\) +\(1 \le W_i \le D_W\) + +The sum of \(N + M\) across all springs is at most 15,000,000. + + +# Input: + +Input begins with an integer \(T\), the number of springs. For each spring there are 11 lines: + +The first line contains the 3 space-separated integers \(N\), \(M\), and \(K\). + +The second line contains the \(K\) space-separated integers \(H_{1..K}\). + +The third line contains the 4 space-separated integers \(A_H\), \(B_H\), \(C_H\), and \(D_H\). + +The fourth line contains the \(K\) space-separated integers \(X_{1..K}\). + +The fifth line contains the 4 space-separated integers \(A_X\), \(B_X\), \(C_X\), and \(D_X\). + +The sixth line contains the \(K\) space-separated integers \(Y_{1..K}\). + +The seventh line contains the 4 space-separated integers \(A_Y\), \(B_Y\), \(C_Y\), and \(D_Y\). + +The eighth line contains the \(K\) space-separated integers \(Z_{1..K}\). + +The ninth line contains the 4 space-separated integers \(A_Z\), \(B_Z\), \(C_Z\), and \(D_Z\). + +The tenth line contains the \(K\) space-separated integers \(W_{1..K}\). + +The eleventh line contains the 4 space-separated integers \(A_W\), \(B_W\), \(C_W\), and \(D_W\). + + +# Output + +For the \(i\)th spring, print a line containing *"Case #i: "*, followed by a single integer, the product of \(S_{1..M}\) modulo 1,000,000,007. + + +# Explanation of Sample + +For the first spring, after the first earthquake, the spring looks as follows (with the 3 cm-long path which a water spider would follow from column 1 to column 2, or vice versa, indicated in red): + +{{PHOTO_ID:825086724953069}} + +After the second earthquake, the spring looks as follows (with the path now only being 2 cm long in either direction): + +{{PHOTO_ID:2854957798069201}} + +Therefore, \(S = [6, 4]\), and the final answer is (6 * 4) modulo 1,000,000,007 = 24. + +For the second spring, \(S = [20, 8, 12]\), with the spring left in the following state after the final earthquake: + +{{PHOTO_ID:2008456099296384}} + +For the third spring, \(S = [222, 382, 110, 266, 136, 98]\). + +For the fourth spring: +- \(H = [37, 11, 18, 6, 34, 33, 14, 20, 13, 5]\) +- \(X = [2, 7, 3, 1, 5, 4, 9, 4, 8, 1, 6, 7]\) +- \(Y = [2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1]\) +- \(Z = [24, 38, 59, 64, 17, 36, 21, 22, 45, 8, 71, 4]\) +- \(W = [16, 9, 55, 35, 58, 9, 4, 77, 55, 27, 10, 29]\) +- \(S = [2926, 3736, 650, 2754, 518, 6108, 5844, 330, 788, 3844, 7096, 9186]\) diff --git a/2020/finals/spider_spring.out b/2020/finals/spider_spring.out new file mode 100644 index 0000000000000000000000000000000000000000..686e4515798d6e35acca100cd04de776323fdbca --- /dev/null +++ b/2020/finals/spider_spring.out @@ -0,0 +1,108 @@ +Case #1: 24 +Case #2: 1920 +Case #3: 633021623 +Case #4: 26104609 +Case #5: 222201367 +Case #6: 294984794 +Case #7: 126230317 +Case #8: 701835095 +Case #9: 476355486 +Case #10: 569072565 +Case #11: 770890219 +Case #12: 331545300 +Case #13: 586465964 +Case #14: 527373588 +Case #15: 660459755 +Case #16: 300060524 +Case #17: 334130994 +Case #18: 388975932 +Case #19: 463042989 +Case #20: 304920201 +Case #21: 113772377 +Case #22: 930176068 +Case #23: 769102540 +Case #24: 408615819 +Case #25: 38456416 +Case #26: 151183167 +Case #27: 495825208 +Case #28: 379977285 +Case #29: 464573744 +Case #30: 54753620 +Case #31: 245195659 +Case #32: 641620440 +Case #33: 598545150 +Case #34: 537132549 +Case #35: 923113036 +Case #36: 526657807 +Case #37: 842401388 +Case #38: 419985955 +Case #39: 964642955 +Case #40: 740556252 +Case #41: 841709531 +Case #42: 878347921 +Case #43: 312086860 +Case #44: 955911834 +Case #45: 276614878 +Case #46: 487022859 +Case #47: 795786176 +Case #48: 343190954 +Case #49: 74385760 +Case #50: 321245598 +Case #51: 965080929 +Case #52: 823067303 +Case #53: 135908834 +Case #54: 585300036 +Case #55: 926495473 +Case #56: 931152045 +Case #57: 894178522 +Case #58: 787600444 +Case #59: 293173412 +Case #60: 68209924 +Case #61: 209830385 +Case #62: 912463885 +Case #63: 575215794 +Case #64: 685242113 +Case #65: 444403624 +Case #66: 962706125 +Case #67: 248763200 +Case #68: 792547224 +Case #69: 91173824 +Case #70: 826460523 +Case #71: 236496818 +Case #72: 562509057 +Case #73: 910930512 +Case #74: 534456170 +Case #75: 3476227 +Case #76: 336398749 +Case #77: 196148390 +Case #78: 715823122 +Case #79: 83714804 +Case #80: 641318271 +Case #81: 725243218 +Case #82: 573492387 +Case #83: 510219501 +Case #84: 199176480 +Case #85: 612804801 +Case #86: 425635646 +Case #87: 969437741 +Case #88: 378662112 +Case #89: 507530203 +Case #90: 513904049 +Case #91: 848838110 +Case #92: 322953305 +Case #93: 517889920 +Case #94: 334400162 +Case #95: 466456064 +Case #96: 798068428 +Case #97: 300601975 +Case #98: 64947010 +Case #99: 468758813 +Case #100: 34487030 +Case #101: 259390400 +Case #102: 457033816 +Case #103: 683157728 +Case #104: 793162616 +Case #105: 992419153 +Case #106: 429921572 +Case #107: 165351954 +Case #108: 782762430 diff --git a/2020/finals/spider_spring_sol.md b/2020/finals/spider_spring_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..c850102eed62e6c5596858c6e26705238ed91dd4 --- /dev/null +++ b/2020/finals/spider_spring_sol.md @@ -0,0 +1,11 @@ +Let a line segment \((a, b)\) refer to a consecutive pair of columns with distinct heights \(a\) and \(b\), such that \(a < b\). If the column indices are \(i\) and \(i+1\), then we'll refer to \(i\) as the line segment's *position*. + +We'll define the *weight* \(w\) of a line segment to be the number of spider paths which would cross it. If its index is \(i\), then \(w = i*(N-i)\). We'll say that each line segment has two separate *endpoints* with values \(a\) and \(b\), each with weight \(w\), and with weighted values \(w*a\) and \(w*b\). + +When evaluating an \(S\) value for a water level \(W\), we'll define a *relevant line segment endpoint* as one with a value greater than \(W\). We'll start by computing the sum of relevant \(b\) endpoints' weighted values, and subtract the sum of relevant \(a\) endpoints' weighted values. There may be line segments for which we only added the \(b\) endpoint but didn't subtract the \(a\) endpoint (due to it being having a value no greater than \(W\)). To account for this, we'll take the sum of relevant \(b\) endpoints' weights, subtract the sum of relevant \(a\) endpoints' weights, multiply that difference by \(W\), and finally subtract that product to arrive at our final answer. + +In order to compute the above values efficiently, we can maintain 4 Fenwick (binary indexed) trees, containing the \(a\) / \(b\) endpoints' weights / weighted values (in each case indexed by the endpoints' values). Assuming these have been maintained, we can compute each \(S\) value in \(O(log(N))\) time. + +At first glance, it appears we'll need to perform up to \(O(NM)\) updates to these trees, given that up to \(N\) line segments may be affected by each of the \(M\) earthquakes. However, we can observe that setting the heights of an interval of columns \([L, R]\) causes all line segments at positions in \([L, R-1]\) to disappear, causes at most 2 new line segments (at positions \(L-1\) and \(R\)) to appear, and doesn't update any other line segments. Therefore, across all of the earthquakes, only \(O(N + M)\) line segments need to be updated, and we just need to not waste time iterating over positions not featuring line segments. + +This can be done either by maintaining a set of intervals of equal-height columns (such that there's a line segment between each consecutive pair of intervals), or by maintaining a segment tree of column heights combined with a set of line segment positions. Either way, these data structures may be updated in amortized \(O(log(N))\) time per earthquake and allow us to only update the Fenwick trees described above for \(O(N + M)\) line segments overall. The total complexity is therefore \(O((N + M) log(N))\). diff --git a/2020/finals/tree_training.cpp b/2020/finals/tree_training.cpp new file mode 100644 index 0000000000000000000000000000000000000000..8b14e90d8e44bbf37566adcb5bcd27924a773c06 --- /dev/null +++ b/2020/finals/tree_training.cpp @@ -0,0 +1,201 @@ +// Tree Training +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) { + char c, r = 0, n = 0; + x = 0; + for (;;) { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define MOD 1000000007 +#define LIM 1000005 +#define LOG 20 + +int N, M; +char S[LIM]; +int Z[LIM]; +int DS[LIM]; + +int TreeCnt(int h) // # of nodes in binary tree with height h +{ + return((1 << (h + 1)) - 1); +} + +// E(z, s, h) = max. *s which can fit alongside z 0s (with s *s before the 1st 0) in a tree of height at most h (only relevant for z > 0) +int E(int z, int s, int h) +{ + if (h >= LOG) + return(N); + int left = min(s, h + 1 - z); // *s at root and along leftmost branch + int cnt = TreeCnt(h); // full tree + cnt -= TreeCnt(h - left); // subtract subtree of 1st 0 + if (cnt >= s) + cnt += TreeCnt(h - left - 1); // add back right subtree of 1st 0 + return(cnt); +} + +// F(p, z, s) = min. tree height for p *s, z 0s, and s *s before 1st 0 (s ignored if z = 0) +int F(int p, int z, int s) +{ + int r1 = z ? (s > 0 ? z : z - 1) : 0; + int r2 = max(r1, min(LOG, p + z - 1)); + while (r1 < r2) + { + int m = (r1 + r2) >> 1; + if ((z ? E(z, s, m) : TreeCnt(m)) >= p) + r2 = m; + else + r1 = m + 1; + } + return(r1); +} + +int ProcessCase() +{ + int i, j, z; + // input + scanf("%s", &S); + N = strlen(S); + // find 0s + M = 0; + Z[M++] = -1; + Fox(i, N) + if (S[i] == '0') + Z[M++] = i; + Z[M++] = N; + // A) z = 0 + int ans = 0; + int v = 0, s = 0; + Fill(DS, 0); + Fox1(i, M - 1) + { + int x = Z[i] - Z[i - 1] - 1; // sequence of x *s before ith 0 + v += x; + DS[2]--, DS[x + 2]++; + } + Fox1(i, N) + { + s += DS[i]; + v += s; + ans = (ans + (LL)v * F(i, 0, 0)) % MOD; + } + // B) 1 <= z <= LOG + Fox1(z, LOG) // z 0s + { + Fox1(i, M - z - 1) // ith 0 as the leftmost 0 + { + int j = i + z - 1; + int c1 = Z[i] - Z[i - 1] - 1; // c1 *s to the left + int c2 = Z[j + 1] - Z[j] - 1; // c2 *s to the right + int m = Z[j] - Z[i] + 1 - z; // m *s in between + Fox(s, c1 + 1) // use s *s to the left + { + int p = s + m, mp = s + m + c2; + int h = F(p, z, s); + while (p <= mp) + { + int p2 = min(mp, E(z, s, h)); + ans = (ans + (LL)(p2 - p + 1) * h) % MOD; + p = p2 + 1, h++; + } + } + } + } + // C) z > LOG + Fox1(i, M - 2) // count contribution of the ith 0 + { + // add # of intervals containing it + ans = (ans + (LL)(Z[i] + 1) * (N - Z[i])) % MOD; + // subtract # of intervals containing it, with <= LOG 0s + FoxI(j, max(1, i - LOG + 1), i) // jth 0 as the leftmost 0 + { + int k = min(M - 1, j + LOG); // first disallowed 0 + int c = (LL)(Z[j] - Z[j - 1]) * (Z[k] - Z[i]) % MOD; + ans = (ans + MOD - c) % MOD; + } + // subtract # of intervals starting with it, with > LOG 0s + int j = i + LOG; + if (j < M) + ans = (ans + MOD - (N - Z[j])) % MOD; + } + return(ans); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/finals/tree_training.in b/2020/finals/tree_training.in new file mode 100644 index 0000000000000000000000000000000000000000..3bed0f45555daaeb33badf1436383ddd780ffe87 --- /dev/null +++ b/2020/finals/tree_training.in @@ -0,0 +1,117 @@ +116 +0 +* +*0 +*** +000 +*0** +0**0*0** +********0*0*****00*00********0 +*000000*0000*0000000000**00**0 +0*0000*0*00000*00***00**0*000*0000*0*000*00000000000*00**0000*0*0*00000*0000*00*00000000*00000*000**000*0000*0**000*000000000*00*0000000*00000*000*0000*00000000*0**00*000*000*00000****000000*0000**000*000000*0000000000000000*00*0**000000*0000000*000000000*000*00**0*00000*0**00000*000***000000*000000 +********0*****0*********0******************************0******************************************0************************************************0*******************************************0*******0******************************0***********************0*********************0**********************0 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+******************************************************************************************************************************************************000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 +000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000****************************************************************************************************************************************************** +*******************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************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**************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************0****************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************0*********************************************************************************************************************************************************************************************************************************0**********************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************0**************************************************************************************************************************************************************************************************************************0************************************************************************************************************************************************************************************************************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+00*0000000*00**0**00**000***00*000**0000*000000000000000*0000**0000*0*00000*0000*000*0*0*0**00*0*00*0*0*000*00*000*000**000*000*00**0*000*00000**0**00*0000000*0*000000**000000**0*000000000**000**00*00*0*00*000**0000*0*000000000*00000000*000*00000*000000*0000**0**00*0*000*00000*000000***0*0**000000*****00000*0000**0***0000*0000000*00*0000000*000 +0*00**00**0*00****0*00*0*0*00000*0****0*0*0******00**000 +*****0******0**0*0**********00****0***********0**0*0*************************000*********************************************0**********0******0*************0***0************0**00***0**** diff --git a/2020/finals/tree_training.md b/2020/finals/tree_training.md new file mode 100644 index 0000000000000000000000000000000000000000..0d4a1caa0f36c93bee55cb7deeb9a33468284e9d --- /dev/null +++ b/2020/finals/tree_training.md @@ -0,0 +1,59 @@ +The great Canadian log drive is [coming to an end](https://www.youtube.com/watch?v=7AheBqBk0iQ&feature=youtu.be&t=1634), for this season at least. Next year, a new crop of log driving hopefuls will compete for their own spot on the river. + +As every Canadian knows, to stay afloat in the forestry industry's competitive job market, you need to familiarize yourself with all kinds of trees. Hoping to land her dream log driving job, Louise was researching online when she stumbled upon an article about [*binary search trees*](https://en.wikipedia.org/wiki/Binary_search_tree) (BSTs). Determined to become a true expert on all there is to know about trees, she decided to continue reading. + +In case you're unfamiliar, Louise learned that a BST is a data structure holding nodes that each store a key (in this case, a non-negative integer). A BST may be empty (that is, holding no nodes), or otherwise consists of a root node that is optionally connected by edges to left and/or right child nodes (which are themselves each the root nodes of "subtree" BSTs, recursively defined in the same fashion). For each node, if its key is \(k\), all keys in its **left** child's subtree (if any) must be **less than or equal** to \(k\), while all keys in its **right** child's subtree (if any) must be **strictly greater** than \(k\). The **height** of a non-empty BST is defined as the **number of edges on the longest path from its root node to any other node** (or 0 if there are no other nodes). + +In practice, BSTs start out empty and are constructed by having nodes inserted into them, one by one. When inserting a new key \(k\) into an existing BST, one compares the root node with \(k\): + +- If the root node doesn't exist, then a node with key \(k\) is created in its place and the insertion is complete. +- Otherwise, if \(k\) is less than or equal to the root node's key, then \(k\) is recursively inserted into the left subtree. +- Otherwise, \(k\) must be greater than the root node's key, in which case it is recursively inserted into the right subtree. + +The following diagrams illustrate the insertion of keys 3, 6, 0, 5, 0, and 3 into an initially empty BST, ending up with a height of 2: + +{{PHOTO_ID:493376864970463}} + +Intrigued, Louise borrowed a data structures textbook from her local library. Upon flipping to the chapter on BSTs, she came upon the following exercise: + +> Given a length-\(N\) string \(S\) in which each character is either `"0"` or `"*"`, please answer the following question for each of the \(N*(N+1)/2\) non-empty substrings of \(S\). + +> For a given substring of length \(x\), choose a sequence \(K_{1..x}\) of integer keys. If the substring's \(i\)th character is `"0"`, then \(K_i = 0\). Otherwise, if it's `"*"`, then \(K_i\) may be chosen to be any positive integer. Suppose the chosen sequence was inserted into an initially empty BST, in order from \(K_1\) to \(K_x\). What is the minimum possible height of the resulting BST? + +For example, if \(S\) = `"0**0*0**"`, then one of the 8*9/2 = 36 substrings for which the above question must be answered is `"**0*0*"`. For this substring, one possible choice of \(K_{1..6}\) would be \([3, 6, 0, 5, 0, 3]\), yielding a BST of height 2 (as illustrated above). This is the minimum height that a BST could have for any possible choice of \(K_{1..6}\) for this substring, meaning that 2 is the answer to the question for this substring. + +Louise would like your help with computing the sum of the answers to all \(N*(N+1)/2\) of these questions. As this sum may be large, you should only compute its value modulo 1,000,000,007. Please help her complete this exercise — the fate of her log driving career rests in your hands! + +# Constraints + +\(1 \le T \le 85\) +\(1 \le N \le 1,000,000\) +\(S_i \in \{``0", ``*"\}\) + +The sum of \(N\) across all exercises is at most 10,000,000. + + +# Input + +Input begins with an integer \(T\), the number of exercises. For each exercise there is a single line consisting of the string \(S_{1..N}\) as specified above. + + +# Output + +For each exercise, print a line containing *"Case #i: "*, followed by a single integer, the sum of the answers to all \(N*(N+1)/2\) questions, modulo 1,000,000,007. + + + +# Explanation of Sample + +In the first exercise, the 3 substrings present are `"*"`, `"0"`, and `"*0"`. The first two each result in a BST with just 1 node (and therefore height 0), while the last one yields a BST with 2 nodes (and therefore height 1). The final answer is therefore \(0 + 0 + 1 = 1\). + +In the second exercise, there are 3 `"*"` substrings, 2 `"**"` substrings, and 1 `"***"` substring. The minimum height of a BST yielded by this last substring is 1 (for example, if \([1, 1, 2]\) is chosen for \(K\)). The final answer is therefore \(0 + 0 + 0 + 1 + 1 + 1 = 3\). + +The third exercise is similar to the second, except that the substring `"000"` must yield a BST with height 2, resulting in an answer of \(0 + 0 + 0 + 1 + 1 + 2 = 4\). + +In the fourth exercise, several of the substrings present are explained below: + +- `"*0*"` can yield a BST with height 1 (for example, with \(K = [1, 0, 2]\)) +- `"0**"` can yield a BST with height 2 (for example, with \(K = [0, 1, 1]\)) +- `"*0**"` can yield a BST with height 2 (for example, with \(K = [1, 0, 1, 2]\)) diff --git a/2020/finals/tree_training.out b/2020/finals/tree_training.out new file mode 100644 index 0000000000000000000000000000000000000000..1169fc2cf1fd86ecbd80b374218c825530f501b6 --- /dev/null +++ b/2020/finals/tree_training.out @@ -0,0 +1,116 @@ +Case #1: 0 +Case #2: 0 +Case #3: 1 +Case #4: 3 +Case #5: 4 +Case #6: 8 +Case #7: 47 +Case #8: 1519 +Case #9: 3691 +Case #10: 3460612 +Case #11: 261512 +Case #12: 2250347 +Case #13: 253533 +Case #14: 4499950 +Case #15: 2316126 +Case #16: 2295141 +Case #17: 704804682 +Case #18: 635473384 +Case #19: 350351725 +Case #20: 385978804 +Case #21: 533983525 +Case #22: 580350160 +Case #23: 991376889 +Case #24: 26575 +Case #25: 82446372 +Case #26: 351416 +Case #27: 1952699 +Case #28: 155906729 +Case #29: 220170 +Case #30: 20590501 +Case #31: 4612530 +Case #32: 301652 +Case #33: 29922051 +Case #34: 182104 +Case #35: 15810162 +Case #36: 3715425 +Case #37: 2549 +Case #38: 75157598 +Case #39: 2682204 +Case #40: 1072445 +Case #41: 1908776 +Case #42: 58324 +Case #43: 16503644 +Case #44: 1677900 +Case #45: 81561785 +Case #46: 5512640 +Case #47: 35612432 +Case #48: 337622 +Case #49: 1686602 +Case #50: 19335 +Case #51: 1054838 +Case #52: 2275280 +Case #53: 8202851 +Case #54: 37974377 +Case #55: 45000512 +Case #56: 3049337 +Case #57: 29161877 +Case #58: 93926525 +Case #59: 101048 +Case #60: 59920466 +Case #61: 912201 +Case #62: 641494 +Case #63: 20188 +Case #64: 97729236 +Case #65: 658149 +Case #66: 2005015 +Case #67: 32085761 +Case #68: 56954680 +Case #69: 13958141 +Case #70: 158793820 +Case #71: 224643 +Case #72: 260351 +Case #73: 9125930 +Case #74: 1204854 +Case #75: 3681603 +Case #76: 22108415 +Case #77: 4438460 +Case #78: 9408602 +Case #79: 9429356 +Case #80: 44598354 +Case #81: 2282400 +Case #82: 1293699 +Case #83: 10158426 +Case #84: 324 +Case #85: 88066313 +Case #86: 44450 +Case #87: 59139787 +Case #88: 17749325 +Case #89: 16888 +Case #90: 61949985 +Case #91: 4274669 +Case #92: 66324486 +Case #93: 46500905 +Case #94: 33025784 +Case #95: 457209 +Case #96: 58845 +Case #97: 5440848 +Case #98: 12611337 +Case #99: 77314509 +Case #100: 215820 +Case #101: 2435161 +Case #102: 55416060 +Case #103: 181418 +Case #104: 52658363 +Case #105: 7773786 +Case #106: 110508996 +Case #107: 2668396 +Case #108: 801430 +Case #109: 3842911 +Case #110: 16852 +Case #111: 2088090 +Case #112: 43486080 +Case #113: 113265282 +Case #114: 4883470 +Case #115: 13868 +Case #116: 121736 diff --git a/2020/finals/tree_training_sol.md b/2020/finals/tree_training_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..58b8272f1acb8a00ebd69b10599fd7c55e318050 --- /dev/null +++ b/2020/finals/tree_training_sol.md @@ -0,0 +1,29 @@ +Let \(E(Z, S, H)\) be the maximum number of `*`s which can fit alongside \(Z\) `0`s, with \(S\) `*`s before the first `0`, in a BST of height at most \(H\) (only relevant for \(Z > 0\)). To compute this in \(O(1)\) time, we should arrange as many `*`s as possible as ancestors of the first `0` (\(\min(S, H-Z+1)\) of them). We can then compute the maximum total number of nodes in a BST of height \(H\) (\(2^{H+1} - 1\)), subtract the number of nodes in the first `0`'s subtree, and potentially add back the number of nodes in the first `0`'s right subtree (if it's possible to get any `*`s there. + +Let \(F(P, Z, S)\) be the minimum BST height for \(P\) `*`s and \(Z\) `0`s, with \(S\) `*`s before the first `0` (again only relevant for \(Z > 0\)). This can be computed in \(O(log(P + Z))\) by simply binary searching for the minimum height and evaluating \(E(Z, S, H)\). + +Finally, let \(G(P)\) be the minimum BST height for \(P\) `*`s and no `0`s. This can similarly be computed in \(O(log(P))\) using binary search or simple math. + +Now, letting \(Z\) be the number of `0`s present in any given substring, we'll we'll separately consider 3 categories of \(Z\) values and add their results up. + +**Category 1:** \(Z = 0\) + +Let \(C_P\) be the number of substrings with \(P\) `*`s and no `0`s. We can compute \(C_{1..N}\) in \(O(N)\) time relatively simply, by iterating through the string and dividing it into intervals of consecutive `*`s (noting that an interval of \(P\) `*`s should increase `C_P` by 1, `C_{P-1}` by 2, and so on, with `C_1` increased by `P`). + +After that, we only need to add up \(C_P * G(P)\) for each \(P\), taking \(O(N log(N))\) (or as little as \(O(N)\)) time. + +**Category 2:** \(1 \le Z \le 20\) + +The significance of the cutoff of \(20\) (\(\lceil log(N) \rceil\)) is explained in Category 3 below. + +For now, for each such value of \(Z\), we can sweep through the string and find the \(O(N)\) possible intervals of `0`s which it could span. Each such interval may be described by the triple \((L, M, R)\), such that there are \(L\) `*`s just left of the leftmost `0`, \(M\) `*`s in total between the leftmost and rightmost `0`s, and \(R\) `*`s just right of the rightmost `0`. + +For each interval, we'll consider each possible number \(S\) of `*`s left of the leftmost 0 to use (such that \(0 \le S \le L\)). Note that this amounts to \(O(N)\) \(S\) values considered across all intervals for any given \(Z\) value. For each \(S\), we need to include \(F(P, Z, S)\) for each \(P\) such that \(S+M \le P \le S+M+R\). This amounts to \(O(N^2)\) \((P, Z, S)\) triples in total, which is too many to individually consider. However, \(F(P, Z, S)\) for fixed values of \(Z\) and \(S\) is monotonic over \(P\) and only takes on \(O(log(N))\) distinct values (dictated by \(E(Z, S, H)\) for \(0 \le H \le \lceil log(N) \rceil\)). These computations can therefore be done in \(O(log(N))\) time for each \(S\). The total time complexity is then \(O(N log(N))\) for each value of \(Z\), and \(O(N log^2(N))\) across all of Category 2. + +**Category 3:** \(Z > 20\) + +For each such \(Z\), there are enough `0`s that the BST's height is entirely dictated by them, no matter how many `*`s there are. In particular, when \(Z > 20\), \(F(P, Z, S)\) equals \(Z-1\) if \(S = 0\), or otherwise equals \(Z\). + +To arrive at the sum of these \(F\) evaluations across all strings with more than 20 `0`s, we'll independently count the "contribution" of each `0` in the string. We'll begin by adding on the total number of intervals containing it (easily computable in \(O(1)\) time), and then subtracting the number of intervals containing it which have no more than 20 `0`s (doable in \(O(log(N))\) time by considering each possible leftmost `0` in such intervals (of which there are at most 21 options). Finally, we'll subtract the number of intervals which begin with it and have more than 20 `0`s (to account for cases in which \(F(P, Z, S)\) equals \(Z-1\) rather than \(Z\)), which is also easily computable in \(O(1)\) time. In total, then, Category 3 can be handled in \(O(N log(N))\). + +Putting all of the above together, the time complexity of this solution comes out to \(O(N log^2(N))\). diff --git a/2020/quals/412634646784856.jpg b/2020/quals/412634646784856.jpg new file mode 100644 index 0000000000000000000000000000000000000000..c12824084697ab59e590a5aca0b491afa5054dc6 --- /dev/null +++ b/2020/quals/412634646784856.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ac14a0abe03a22d80a180d80ca2a2875fcb37a5b2ff62215aca9c4b1069f7cc9 +size 8849 diff --git a/2020/quals/781032149369958.jpg b/2020/quals/781032149369958.jpg new file mode 100644 index 0000000000000000000000000000000000000000..0000d002a9ea85000e2a19153cda3b804c39d72f --- /dev/null +++ b/2020/quals/781032149369958.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1631cdf7bee95e6dff29ee49e8fba4e06ae679c745662a6a382fc8f34436a8cb +size 6403 diff --git a/2020/quals/alchemy.cpp b/2020/quals/alchemy.cpp new file mode 100644 index 0000000000000000000000000000000000000000..3f8809c34f0c9fe6575b10d75d4899fe0db7a4be --- /dev/null +++ b/2020/quals/alchemy.cpp @@ -0,0 +1,104 @@ +// Alchemy +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) { + char c, r = 0, n = 0; + x = 0; + for (;;) { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 100001 + +int N; +char P[LIM]; + +bool ProcessCase() +{ + // input + Read(N); + scanf("%s", P); + // count difference between counts of As and Bs + int i, c = 0; + Fox(i, N) + c += (P[i] == 'A' ? 1 : -1); + // absolute difference must be equal to 1 + return(Abs(c) == 1); +} + +int main() { + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %c\n", t, ProcessCase() ? 'Y' : 'N'); + return(0); +} \ No newline at end of file diff --git a/2020/quals/alchemy.in b/2020/quals/alchemy.in new file mode 100644 index 0000000000000000000000000000000000000000..437447bfe8ce38e050391eecb1843125b4430afd --- /dev/null +++ b/2020/quals/alchemy.in @@ -0,0 +1,249 @@ +124 +11 +AAAAAAAAAAA +11 +BAAAAAAAAAA +11 +BBAAAAAAAAA +11 +BBBAAAAAAAA +11 +BBBBAAAAAAA +11 +BBBBBAAAAAA +11 +BBBBBBAAAAA +11 +BBBBBBBAAAA +11 +BBBBBBBBAAA +11 +BBBBBBBBBAA +11 +BBBBBBBBBBA +11 +BBBBBBBBBBB +3 +BAB +3 +BBB +5 +AABBA +7 +BAAABAA +11 +BBBAABAAAAB +11 +BABBBABBABB +21 +AAABBABBBBBABBAAAABAB +21 +AAAAAAAAAAAAAAAAAAAAA +51 +BBBBBBBBBBBBAAAAABBAAAAAAAAAAAAABBBBAAAAAAABBBBBBBB +51 +BBAAAABBBAABBBBBBAABBBABBBAAAAAABBBAABABABBAABAABBA +99999 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The Philosopher's Stone (also known as the Red Stone, the Fifth Element, and the Elixir of Life in liquid form) is a powerful alchemical substance rumoured to support the transmutation of silver, gold, and even human lives. Edgar is hoping to use the stone to restore his brother's body to its original form following a terrifying human transmutation accident. + +It is known that Philosopher's Stones break apart into an odd number of shards when they're used up. After searching far and wide, Edgar has collected \(N\) shards of what he believes to be a previously-used Philosopher's Stone, where \(N\) is an odd integer. He believes that fusing these shards back together will yield a real stone in its entirety. + +Leveraging his encyclopedic knowledge of alchemy, Edgar recalls an ancient book with the following ritual for fusing shards: + + 1. Arrange the \(N\) shards in a row, numbered from 1 to \(N\). + 2. Note the colour \(C_i\) of each shard \(i\), which is either *auburn* or *black* (denoted by the characters *"A"* and *"B"* respectively). + 3. Choose a consecutive sequence of three shards, remove them from the row, and fuse them together into one new shard. The three shards must not all be the same colour, or such an intense concentration of similar energies will cause a dangerous rebound explosion. Instead, there must be two differently-coloured shards that will neutralize each other, get converted into energy (via the Law of Equivalent Exchange), and become infused with the third shard. In other words, assuming no explosion, the resulting shard's colour will be equal to the most common colour present amongst the three chosen shards. + 4. Place the resulting shard back into the original gap in the row, with the remaining shards in the same relative order before and after it. + 5. Repeat steps 3 and 4 until only one shard remains. + +Note that each time steps 3 and 4 are performed, the number of remaining shards decreases by 2. Therefore, given that the initial number of shards is odd, enough successful repetitions will result in exactly one final shard: the Philosopher's Stone. Help Edgar determine whether or not it's possible to get to that point and hopefully save Alfred, without causing any rebound explosions along the way. + + +# Input + +Input begins with an integer \(T\), the number of Philosopher's Stones that Edgar tries to reconstitute. For each stone there are two lines. The first contains the integer \(N\). The second contains the length-\(N\) string \(C_{1..N}\). + +# Output + +For the \(i\)th stone, output a line containing *"Case #i: "* followed by a single character, either *"Y"* if it's possible For Edgar to end up with the Philosopher's Stone, or *"N"* if not. + +# Constraints + +\(1 \le T \le 95\) +\(3 \le N \le 99,999\) +\(N\) is odd +\(C_i \in \{A, B\}\) + +The sum of \(N\) over all test cases is at most 1,000,000. + +# Explanation of Sample + +In the first case, Edgar is able to fuse the three shards together, as they include two differently-coloured shards, yielding a single black-coloured shard: the Philosopher's Stone. + +In the second case, the three black shards will inevitably cause a rebound explosion if fused. + +In the third case, Edgar can begin by fusing the first three shards into a single auburn-coloured shard. The row of shards will then feature colours *"ABA"*, and those shards can be fused into the Philosopher's Stone. + diff --git a/2020/quals/alchemy.out b/2020/quals/alchemy.out new file mode 100644 index 0000000000000000000000000000000000000000..d2888c65dfb48cdb282fcd8a14d8de8683b1b717 --- /dev/null +++ b/2020/quals/alchemy.out @@ -0,0 +1,124 @@ +Case #1: N +Case #2: N +Case #3: N +Case #4: N +Case #5: N +Case #6: Y +Case #7: Y +Case #8: N +Case #9: N +Case #10: N +Case #11: N +Case #12: N +Case #13: Y +Case #14: N +Case #15: Y +Case #16: N +Case #17: Y +Case #18: N +Case #19: Y +Case #20: N +Case #21: Y +Case #22: N +Case #23: Y +Case #24: N +Case #25: Y +Case #26: N +Case #27: Y +Case #28: Y +Case #29: Y +Case #30: N +Case #31: Y +Case #32: N +Case #33: Y +Case #34: N +Case #35: Y +Case #36: N +Case #37: Y +Case #38: Y +Case #39: N +Case #40: N +Case #41: Y +Case #42: N +Case #43: N +Case #44: N +Case #45: Y +Case #46: N +Case #47: Y +Case #48: Y +Case #49: N +Case #50: N +Case #51: Y +Case #52: N +Case #53: Y +Case #54: Y +Case #55: N +Case #56: Y +Case #57: N +Case #58: Y +Case #59: N +Case #60: N +Case #61: Y +Case #62: N +Case #63: N +Case #64: Y +Case #65: Y +Case #66: Y +Case #67: Y +Case #68: Y +Case #69: Y +Case #70: Y +Case #71: N +Case #72: Y +Case #73: N +Case #74: Y +Case #75: N +Case #76: N +Case #77: Y +Case #78: N +Case #79: Y +Case #80: Y +Case #81: N +Case #82: N +Case #83: Y +Case #84: N +Case #85: Y +Case #86: Y +Case #87: N +Case #88: N +Case #89: N +Case #90: N +Case #91: Y +Case #92: N +Case #93: Y +Case #94: Y +Case #95: Y +Case #96: N +Case #97: N +Case #98: Y +Case #99: Y +Case #100: N +Case #101: N +Case #102: N +Case #103: Y +Case #104: N +Case #105: N +Case #106: N +Case #107: N +Case #108: N +Case #109: Y +Case #110: Y +Case #111: Y +Case #112: N +Case #113: Y +Case #114: Y +Case #115: Y +Case #116: N +Case #117: N +Case #118: Y +Case #119: Y +Case #120: Y +Case #121: Y +Case #122: Y +Case #123: Y +Case #124: N diff --git a/2020/quals/alchemy_sol.md b/2020/quals/alchemy_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..53209c16b670750a35cd96783f7a859d62cc0371 --- /dev/null +++ b/2020/quals/alchemy_sol.md @@ -0,0 +1,7 @@ +Let \(F_A\) and \(F_B\) be the number of *"A"* (auburn) shards and the number of *"B"* (black) shards, respectively. + +We can observe that every time shards are fused \(F_A\) and \(F_B\) each decrease by \(1\). One implication of this is that total number of shards \(F_A + F_B\) decreases by \(2\) each time. Since \(N\) is initially a positive odd integer, this means that the sequence of shards will always be reduced to a single shard after enough fusing operations, as long as they can all be performed safely. A more interesting implication is that \(F_A - F_B\) remains constant between operations. Since \(F_A - F_B\) must be equal to either \(1\) or \(-1\) when there's one shard remaining, we must also have \(|F_A - F_B| = 1\) in the initial sequence in order for that final state to be achievable. + +So far, we've shown that the answer must be *"N"* if \(|F_A - F_B| \neq 1\). The remaining question is whether the answer is always *"Y"* if \(|F_A - F_B| = 1\). The only way it could be *"N"* is if it were impossible to perform a fusing operation at any point while there are three or more shards. This is only impossible if all shards are the same colour (as there must otherwise be at least one consecutive triple of shards not all sharing the same colour), which cannot ever be the case if \(N \geq 3\) and \(|F_A - F_B| = 1\) (as we must have \(F_A \geq 1\) and \(F_B \geq 1\)). + +Therefore, we have only to loop over the shards in \(O(N)\) time to compute \(F_A\) and \(F_B\), and then check whether or not \(|F_A - F_B| = 1\). diff --git a/2020/quals/running_on_fumes_ch1.cpp b/2020/quals/running_on_fumes_ch1.cpp new file mode 100644 index 0000000000000000000000000000000000000000..485f9d5674569178b87f4e706c7cce37af175977 --- /dev/null +++ b/2020/quals/running_on_fumes_ch1.cpp @@ -0,0 +1,127 @@ +// Running on Fumes - Chapter 1 +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 1000008 +#define IMP 1e18 + +int N, M; +int C[LIM]; +deque> D; // + +LL ProcessCase() +{ + int i; + // input + Read(N), Read(M); + Fox(i, N) + Read(C[i]); + // iterate over cities while maintaining deque of min. costs + D.clear(); + D.push_back(mp(0, 0)); + Fox(i, N) + { + // remove entries too far back to reach current city + while (!D.empty() && D.front().x < i - M) + D.pop_front(); + // impossible to make it this far? + if (D.empty()) + return(-1); + // consider refueling here + if (C[i]) + { + LL d = D.front().y + C[i]; + // remove no-longer-relevant entries + while (!D.empty() && d <= D.back().y) + D.pop_back(); + D.push_back(mp(i, d)); + } + } + return(D.front().y); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %lld\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/quals/running_on_fumes_ch1.in b/2020/quals/running_on_fumes_ch1.in new file mode 100644 index 0000000000000000000000000000000000000000..83d10e4f56a31c8d413a149702da1ed50ac91079 --- /dev/null +++ b/2020/quals/running_on_fumes_ch1.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:913a54b16b85db350fbf74508521343c8bea6cf22614a6fac4a4e7384d34e8ad +size 10866723 diff --git a/2020/quals/running_on_fumes_ch1.md b/2020/quals/running_on_fumes_ch1.md new file mode 100644 index 0000000000000000000000000000000000000000..0e76e20571002e85f45506a00c0da065c2b5104b --- /dev/null +++ b/2020/quals/running_on_fumes_ch1.md @@ -0,0 +1,37 @@ +**Note: This problem shares similarities with [Chapter 2](https://www.facebook.com/codingcompetitions/hacker-cup/2020/qualification-round/problems/D2). The solution to either chapter may help with solving the other, so please consider reading both first.** + +You just landed yourself a gig as a delivery driver for a nationwide supply chain. You've been assigned a series of long-haul jobs, so it's time to get to work. + +For each job, you will be provided with a map of the relevant region, which includes \(N\) cities (numbered from 1 to \(N\)) and \(N-1\) two-way roads running amongst them. **The cities are connected by roads in a single line**, such that there's a road between each pair of consecutive cities. In other words, cities \(i\) and \(j\) are directly connected by a road if and only if \(|i - j| = 1\). + +You will begin in city \(1\) with a shipment of supplies to be delivered to city \(N\), and with a full gas tank having a capacity of \(M\) gallons. You will then have two options at each point in time: + + 1. Drive along a road from your current city to an adjacent one, using up 1 gallon of gas. You may not do this if your tank is empty, but it's fine if your tank becomes empty as a result. + 2. Fill your tank all the way back up to \(M\) gallons of gas at a cost of \(C_i\) dollars, where \(i\) is your current city. Note that the cost is independent of how much gas your tank had before refueling. You may not do this if city \(i\) has no gas station (indicated with \(C_i = 0\)). + +Determine the minimum cost required to arrive at city \(N\), if it's possible at all. + +# Input + +Input begins with an integer \(T\), the number of long-haul jobs you've been assigned. For each job there is first a line containing the space-separated integers \(N\) and \(M\). Then, \(N\) lines follow, the \(i\)th of which contains the single integer \(C_i\). + +# Output + +For the \(i\)th job, output a line containing *"Case #i: "* followed by a single integer, the minimum cost in dollars to get from city 1 to city \(N\), or \(-1\) if it's impossible. + +# Constraints + +\(1 \le T \le 85\) +\(2 \le N \le 1,000,000\) +\(1 \le M \le N\) +\(0 \le C_i \le 1,000,000,000\) + +The sum of \(N\) across all jobs is at most 4,000,000. + +# Sample Explanation + +In the first job, you will begin in city 1 with 3 gallons of gas. You cannot drive all the way to your destination (city 5) without refueling along the way, as that would require a total of 4 gallons of gas. The cheapest option is to drive to city 2, top up your tank for a cost of $20, and then drive through cities 3 and 4 before reaching city 5 with no more gas to spare. + +In the second job, your gas tank only has a capacity of 2 gallons. In this case, the cheapest strategy involves depleting all of your gas to drive to city 3, refueling there for $30, and then depleting all of your gas to reach city 5. + +In the third job, your gas tank only has a capacity of 1 gallon. No matter what you do, you will deplete all of your gas in the drive from city 3 to city 4, where there will be no way to refuel to reach city 5. diff --git a/2020/quals/running_on_fumes_ch1.out b/2020/quals/running_on_fumes_ch1.out new file mode 100644 index 0000000000000000000000000000000000000000..033010f8346c058c024738dc2eaadcfba6d56f2e --- /dev/null +++ b/2020/quals/running_on_fumes_ch1.out @@ -0,0 +1,116 @@ +Case #1: 20 +Case #2: 30 +Case #3: -1 +Case #4: 165 +Case #5: 0 +Case #6: 50 +Case #7: 19 +Case #8: 0 +Case #9: -1 +Case #10: 2999999979 +Case #11: 3333744 +Case #12: 627743982 +Case #13: 1041706719 +Case #14: 135999268450 +Case #15: 25350250167 +Case #16: 10259 +Case #17: 258 +Case #18: 2 +Case #19: 2 +Case #20: 1 +Case #21: 2998642649997 +Case #22: 8 +Case #23: 2 +Case #24: 8 +Case #25: 3 +Case #26: 16 +Case #27: 14 +Case #28: 22 +Case #29: 12 +Case #30: 11 +Case #31: 13 +Case #32: 11 +Case #33: 12 +Case #34: -1 +Case #35: 6 +Case #36: -1 +Case #37: -1 +Case #38: 0 +Case #39: 3 +Case #40: 2 +Case #41: 3 +Case #42: 2 +Case #43: 4 +Case #44: 3 +Case #45: 3 +Case #46: 12 +Case #47: 19 +Case #48: 15 +Case #49: 11 +Case #50: 13 +Case #51: 12 +Case #52: 14 +Case #53: 13 +Case #54: -1 +Case #55: -1 +Case #56: -1 +Case #57: 1491 +Case #58: 0 +Case #59: 3 +Case #60: 3 +Case #61: 3 +Case #62: 3 +Case #63: 3 +Case #64: 3 +Case #65: 2 +Case #66: 16 +Case #67: 0 +Case #68: 11 +Case #69: 12 +Case #70: 12 +Case #71: 21 +Case #72: 10 +Case #73: 10 +Case #74: -1 +Case #75: -1 +Case #76: -1 +Case #77: -1 +Case #78: 0 +Case #79: 3 +Case #80: 3 +Case #81: 3 +Case #82: 3 +Case #83: 3 +Case #84: 2 +Case #85: 3 +Case #86: 16 +Case #87: 10 +Case #88: 12 +Case #89: 11 +Case #90: 19 +Case #91: 24 +Case #92: 17 +Case #93: 18 +Case #94: -1 +Case #95: 644 +Case #96: -1 +Case #97: -1 +Case #98: 0 +Case #99: 3 +Case #100: 3 +Case #101: 3 +Case #102: 3 +Case #103: 3 +Case #104: 3 +Case #105: 3 +Case #106: 14 +Case #107: 26 +Case #108: 17 +Case #109: 20 +Case #110: 12 +Case #111: 10 +Case #112: 21 +Case #113: 11 +Case #114: -1 +Case #115: -1 +Case #116: -1 diff --git a/2020/quals/running_on_fumes_ch1_sol.md b/2020/quals/running_on_fumes_ch1_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..2c83e00d9f2bdd73db00cb187ad6c475d663e74b --- /dev/null +++ b/2020/quals/running_on_fumes_ch1_sol.md @@ -0,0 +1,13 @@ +We first observe that it's never beneficial to move back to a lower-indexed city — we'll only move forwards through the \(N\) cities in order. + +Let \(F_i\) be the minimum cost required to be at city \(i\) with a full tank of gas, and \(G_i\) be the minimum cost required to arrive at city \(i\) at all (with each equal to \(\infty\) if unachievable). Our answer will be \(G_N\). + +We know that \(F_1 = G_1 = 0\). For each \(i > 1\), \(G_i\) is equal to the minimum value of \(F_j\) across all cities \(j\) such that it's possible to drive directly from \(j\) to \(i\) on a single tank of gas (in other words, such that \(max(i-M,1) \le j < i\)). \(F_i\) is then simply equal to \(G_i + C_i\) if \(C_i \ne 0\), or is equal to \(\infty\) otherwise. + +Since each \(G\) and \(F\) value only depends on previous ones, this suggests an approach involving iterating over the cities in order from \(1\) to \(N\) and simply computing their \(G\) and \(F\) values along the way. However, a direct implementation of the above would be too inefficient, involving considering up to \(M\) prior \(F\) values in order to compute each \(G\) value and yielding a time complexity of \(O(N*M)\). + +We can improve on this through observations around which \(F\) values may be relevant when computing subsequent \(G\) values. If there exists a pair of cities \(i\) and \(j\) such that \(i < j\) and \(F_i \ge F_j\), then we can disregard \(F_i\) going forward, as \(F_j\) will always be at least as beneficial. + +This suggests that we can maintain an ordered list \(L\) of cities which may ever be relevant, which have both increasing indices and increasing \(F\) values. As we iterate over the \(N\) cities, we should remove the earliest cities from \(L\) once they become too far back to still be used for the current city's \(G\) value, we should append new cities onto the end of \(L\) when we compute their \(F\) values, and right before appending each such city, we should remove cities from \(L\) which will never be relevant anymore (due to having an \(F\) value larger than or equal to the new city). We can then compute each current city's \(G\) value in just \(O(1)\) time (by using the \(F\) value of the earliest city in \(L\), which must be the minimum one). Given the operations involved, representing \(L\) as a deque is most convenient. + +Every city will be added to and removed from \(L\) at most once, and we'll perform only a constant number of other operations while computing each city's \(F\) and \(G\) values. Therefore, the time complexity of this solution is \(O(N)\). diff --git a/2020/quals/running_on_fumes_ch2.cpp b/2020/quals/running_on_fumes_ch2.cpp new file mode 100644 index 0000000000000000000000000000000000000000..0f3343115de443ea8f22c181d88ff1eb11c50285 --- /dev/null +++ b/2020/quals/running_on_fumes_ch2.cpp @@ -0,0 +1,198 @@ +// Running on Fumes - Chapter 2 +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 1000008 +#define LIM2 2100000 + +int N, M, A, B; +vector con[LIM]; +int P[LIM], C[LIM]; +int mZ, mD, mHas[LIM], mC[LIM]; +int sz; +LL tree[LIM2]; + +// DFS from node i, while populating parents +void rec1(int i) +{ + int j, c; + Fox(j, Sz(con[i])) + if ((c = con[i][j]) != P[i]) + { + P[c] = i; + rec1(c); + } +} + +// DFS from node i, with depth d and previous node p, while populating min. C values per depth +void rec2(int i, int d, int p) +{ + int j, c; + if (C[i]) + { + Max(mD, d); + if (mHas[d] == mZ) + Min(mC[d], C[i]); + else + mHas[d] = mZ, mC[d] = C[i]; + } + Fox(j, Sz(con[i])) + if ((c = con[i][j]) != P[i]) + if (c != p) + rec2(c, d + 1, p); +} + +// update value at index i in segment tree to be at most v +void Update(int i, LL v) +{ + i += sz; + while (i) + { + Min(tree[i], v); + i >>= 1; + } +} + +// query min. value in segment tree within indices [a, b] +LL Query(int i, int r1, int r2, int a, int b) +{ + if (a <= r1 && r2 <= b) + return(tree[i]); + i <<= 1; + int m = (r1 + r2) >> 1; + LL ret = (LL)INF * INF; + if (a <= m) + Min(ret, Query(i, r1, m, a, b)); + if (b > m) + Min(ret, Query(i + 1, m + 1, r2, a, b)); + return(ret); +} + +LL ProcessCase() +{ + int i, j, k, p, d; + // input + Read(N), Read(M), Read(A), Read(B); + Fox1(i, N) + { + Read(j), Read(C[i]); + if (j) + { + con[i].pb(j); + con[j].pb(i); + } + } + // root tree at B + P[B] = -1; + rec1(B); + // init segment tree of min. costs + Fill(tree, 60); + for (sz = 1; sz < N; sz <<= 1); + // iterate along path from A to B while maintaining segment tree + Update(0, 0); + for (p = A, i = P[A], k = 1; i != B; p = i, i = P[i], k++) + { + // explore branches off of main path + mZ++, mD = -1; + rec2(i, 0, p); + // consider refueling here or within an attached branch + Fox(d, mD + 1) + if (k - d >= 0 && mHas[d] == mZ) + { + LL m = Query(1, 0, sz - 1, max(0, k - (M - d)), k); + Update(k - d, m + mC[d]); + } + } + // clear graph + Fox1(i, N) + con[i].clear(); + // get answer + LL ans = Query(1, 0, sz - 1, max(0, k - M), k - 1); + return ans >= (LL)INF * INF ? -1 : ans; +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %lld\n", t, ProcessCase()); + return(0); +} diff --git a/2020/quals/running_on_fumes_ch2.in b/2020/quals/running_on_fumes_ch2.in new file mode 100644 index 0000000000000000000000000000000000000000..1577ade3d3decd6bf078e731900be01b2d2efe95 --- /dev/null +++ b/2020/quals/running_on_fumes_ch2.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8180aa82db1150255ed85379ec7685a73e3d1ec3d6ebef6ab822c2bf1e987b5e +size 24868533 diff --git a/2020/quals/running_on_fumes_ch2.md b/2020/quals/running_on_fumes_ch2.md new file mode 100644 index 0000000000000000000000000000000000000000..e2ab274420a4fd835570600febc07c318a3d463f --- /dev/null +++ b/2020/quals/running_on_fumes_ch2.md @@ -0,0 +1,50 @@ +**Note: This problem shares similarities with [Chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2020/qualification-round/problems/D1). The solution to either chapter may help with solving the other, so please consider reading both first.** + +After working for many months as a delivery driver for a nationwide supply chain, you've finally received a promotion! This promotion has come with a nice pay bump, but also with the responsibility of serving regions with more complex maps. With no opportunity for a vacation, you've promptly been assigned a new series of long-haul jobs. + +For each job, you will be provided with a map of the relevant region, which includes \(N\) cities (numbered from \(1\) to \(N\)) and \(N-1\) two-way roads running amongst them. In computer science terms, **the map is organized like a tree rooted at city 1**, such each city \(i\) is connected to its parent city \(P_i\) by a road (aside from city 1, which has no parent, with \(P_1\) given as 0). It's possible to drive from any city to any other city by following a sequence of roads. Note that each road may be driven along in either direction (either from city \(i\) to city \(P_i\), or vice versa). + +You will begin in a given city \(A\) with a shipment of supplies to be delivered to another given city \(B\), and with a full gas tank having a capacity of \(M\) gallons. You will then have two options at each point in time: + + 1. Drive along a road from your current city to an adjacent one, using up 1 gallon of gas. You may not do this if your tank is empty, but it's fine if your tank becomes empty as a result. + 2. Fill your tank all the way back up to \(M\) gallons of gas at a cost of \(C_i\) dollars, where \(i\) is your current city. Note that the cost is independent of how much gas your tank had before refueling. You may not do this if city \(i\) has no gas station (indicated with \(C_i\) = 0). + +Determine the minimum cost required to complete your delivery, if it's possible at all. + +# Input + +Input begins with an integer \(T\), the number of long-haul jobs you are assigned. For each job there is first a line containing the space-separated integers \(N\), \(M\), \(A\), and \(B\). Then, \(N\) lines follow, the \(i\)th of which contains the space-separated integers \(P_i\) and \(C_i\). + +# Output + +For the \(i\)th job, output a line containing *"Case #i: "* followed by a single integer, the minimum cost in dollars to get from city \(A\) to city \(B\), or \(-1\) if it's impossible. + +# Constraints + +\(1 \le T \le 80\) +\(2 \le N \le 1,000,000\) +\(1 \le M \le N\) +\(1 \le A, B \le N\) +\(A \ne B\) +\(0 \le P_i \le N\) +\(0 \le C_i \le 1,000,000,000\) + +The sum of \(N\) across all jobs is at most 3,000,000. + +# Explanation of Sample + +The regional map for the first job looks as follows: + +{{PHOTO_ID:781032149369958}} + +You will begin in city 2 with 1 gallon of gas. The cheapest strategy involves driving to city 4, refueling there for a cost of $40, and then driving onwards to city 3. + +The regional map for the next 3 jobs looks as follows, such that you need to drive from city 5 to city 3: + +{{PHOTO_ID:412634646784856}} + +In the second job, your gas tank has a capacity of 1 gallon. No matter what you do, you will deplete all of your gas in the drive from city 8 to city 6, where there will be no way to refuel and proceed to eventually reach city 3. + +In the third job, your gas tank has a capacity of 2 gallons. The cheapest strategy involves refueling at both cities 8 and 1 on the way to city 3, costing a total of $30 + $30 = $60. + +In the fourth job, your gas tank has a capacity of 3 gallons. This isn't quite enough to drive all the way to city 3 without refueling. The cheapest strategy involves depleting all of your gas to drive through cities 8 and 6 to city 7, refueling there for $20, and then depleting all of your gas to drive through cities 6 and 1 to reach city 3. diff --git a/2020/quals/running_on_fumes_ch2.out b/2020/quals/running_on_fumes_ch2.out new file mode 100644 index 0000000000000000000000000000000000000000..7c2f4f88b61ec33538c6af26a64c93a8a32ec882 --- /dev/null +++ b/2020/quals/running_on_fumes_ch2.out @@ -0,0 +1,116 @@ +Case #1: 40 +Case #2: -1 +Case #3: 60 +Case #4: 20 +Case #5: 104 +Case #6: 17 +Case #7: 19 +Case #8: 0 +Case #9: 2999999982 +Case #10: 5 +Case #11: 210 +Case #12: -1 +Case #13: -1 +Case #14: 525 +Case #15: 410 +Case #16: 2165 +Case #17: 4 +Case #18: 74626801 +Case #19: 10 +Case #20: 1175 +Case #21: 20556 +Case #22: 11 +Case #23: 18 +Case #24: 1 +Case #25: 1034 +Case #26: 269 +Case #27: 31 +Case #28: 30 +Case #29: 876 +Case #30: 0 +Case #31: 4 +Case #32: 2983 +Case #33: 171 +Case #34: 8 +Case #35: 46 +Case #36: 1565 +Case #37: -1 +Case #38: 0 +Case #39: 28 +Case #40: 20 +Case #41: -1 +Case #42: 8351 +Case #43: 89 +Case #44: 7 +Case #45: 0 +Case #46: 0 +Case #47: -1 +Case #48: 20 +Case #49: 8 +Case #50: 2818 +Case #51: 10 +Case #52: 52 +Case #53: -1 +Case #54: 0 +Case #55: 66 +Case #56: 856 +Case #57: 99 +Case #58: 22 +Case #59: 6 +Case #60: 0 +Case #61: 107 +Case #62: 3302 +Case #63: 242 +Case #64: 17 +Case #65: 798 +Case #66: 5 +Case #67: 2130 +Case #68: 1716 +Case #69: 61 +Case #70: -1 +Case #71: 0 +Case #72: 160 +Case #73: 158 +Case #74: 2723 +Case #75: -1 +Case #76: 11 +Case #77: 8268 +Case #78: 31 +Case #79: -1 +Case #80: 0 +Case #81: 0 +Case #82: 4159 +Case #83: 274 +Case #84: 5 +Case #85: 2 +Case #86: 3019 +Case #87: 1 +Case #88: 0 +Case #89: 51 +Case #90: 2 +Case #91: 8 +Case #92: 17 +Case #93: 0 +Case #94: 0 +Case #95: 7 +Case #96: 0 +Case #97: 445 +Case #98: 0 +Case #99: 0 +Case #100: 4327 +Case #101: 0 +Case #102: -1 +Case #103: 13 +Case #104: 1548 +Case #105: 23956 +Case #106: 1 +Case #107: 0 +Case #108: 8 +Case #109: 10 +Case #110: -1 +Case #111: 209 +Case #112: 0 +Case #113: 386 +Case #114: 52 +Case #115: 0 +Case #116: 0 diff --git a/2020/quals/timber.cpp b/2020/quals/timber.cpp new file mode 100644 index 0000000000000000000000000000000000000000..9ec61911acacbd0144603370e23fa3b0095e5107 --- /dev/null +++ b/2020/quals/timber.cpp @@ -0,0 +1,113 @@ +// Timber +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) { + char c, r = 0, n = 0; + x = 0; + for (;;) { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 800005 + +int N; +PR P[LIM]; +unordered_map M[2]; // [left-to-right, right-to-left] + +int ProcessCase() +{ + int i, d, ans = 0; + Read(N); + Fox(i, N) + Read(P[i].x), Read(P[i].y); + sort(P, P + N); + Fox(d, 2) + M[d].clear(); + Fox(d, 2) + { + if (d == 1) + reverse(P, P + N); + Fox(i, N) + Max(M[d][P[i].x + P[i].y * (d ? -1 : 1)], M[d][P[i].x] + P[i].y); + for (auto I : M[d]) + Max(ans, I.y + M[1 - d][I.x]); + } + return(ans); +} + +int main() { + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/quals/timber.in b/2020/quals/timber.in new file mode 100644 index 0000000000000000000000000000000000000000..d2755e9606bcbdffded759b185d6d7aef1c670e7 --- /dev/null +++ b/2020/quals/timber.in @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bd658bca70a2c15e1642d19004a326d8646839ed5e1daeeeb93e1c46d0bb3bd2 +size 30148379 diff --git a/2020/quals/timber.md b/2020/quals/timber.md new file mode 100644 index 0000000000000000000000000000000000000000..2fde82785800605ae0d55007a9748c274c5ac899 --- /dev/null +++ b/2020/quals/timber.md @@ -0,0 +1,35 @@ +Every year, millions of logs are sent floating downstream in [the great Canadian log drive](https://www.youtube.com/watch?v=7AheBqBk0iQ). But as every Canadian knows, you can't have a successful log drive without logs. The first step in the logging season is to fell a great number of trees, taking care not to pile the trees up in a way that will make them hard to move. + +Eva has been assigned a section of forest with \(N\) trees standing in a row. Looking at them from the side and modeling the ground as a number line, the \(i\)th tree is at position \(P_i\) (measured in metres) and has a height of \(H_i\) metres. All of the trees are at distinct positions. + +Each tree may be cut down such that it falls either to the left or to the right, or it may be left standing (it's important to farm trees sustainably). If the \(i\)th tree is cut down to the left, it will form a **timber interval** of \([P_i - H_i, P_i]\) on the ground, while if it is cut down to the right, it will form a timber interval \([P_i, P_i + H_i]\). + +The trees that are cut down will be moved into a nearby river to begin their long journey to the mill. It's hard to move a tree that has another tree on top of it, so Eva will not cut down trees in a way that would cause the intersection of any pair of timber intervals to have positive length. It's fine however, and indeed desirable, for two timber intervals to share an endpoint in common. For example, one tree may be cut down to create a timber interval of \([a, b]\) and another may be cut down to create a timber interval of \([b, c]\). It's also fine for timber intervals to overlap with trees which have not been cut down. Uncut trees are essentially ignored. + +A **combined timber interval** is an interval where each point within it is within at least one timber interval. For example, if you have timber intervals \([a, b]\) and \([b, c]\) (for any \(a < b < c\)), then \([a, c]\) is a combined timber interval. + +Eva would like to form the largest combined timber interval possible, as that represents a series of trees that can all be pushed into the river at once. Determine the maximum possible length that a combined timber interval could have if Eva cuts down trees optimally. + +# Input + +Input begins with an integer \(T\), the number of forest sections that Eva will work in. For each section, there is first a line containing the single integer \(N\). Then, \(N\) lines follow, the \(i\)th of which contains the space-separated integers \(P_i\) and \(H_i\). + +# Output + +For the \(i\)th section, print a line containing *"Case #i: "* followed by a single integer, the length of the largest possible combined timber interval that Eva can create in that section of forest. + +# Constraints +\(1 \le T \le 90\) +\(1 \le N \le 800,000\) +\(-500,000,000 \le P_i \le 500,000,000\) +\(1 \le H_i \le 500,000,000\) + +The sum of \(N\) across all forest sections is at most 4,000,000. + +# Explanation of Sample + +In the first case, if both trees are cut down such that they fall to the right, a combined timber interval of length 9 (between positions 0 and 9) will be formed. + +In the second case, if the first tree falls to the right and the second falls to the left, they will similarly form a combined timber interval of length 9. + +In the third case, no matter how the trees are cut down, their endpoints cannot coincide. Therefore, the longest possible combined timber interval consists of just the third tree being chopped down. diff --git a/2020/quals/timber.out b/2020/quals/timber.out new file mode 100644 index 0000000000000000000000000000000000000000..1c38ce326c23e9c4a8d0e1769897d834edaf6a34 --- /dev/null +++ b/2020/quals/timber.out @@ -0,0 +1,113 @@ +Case #1: 9 +Case #2: 9 +Case #3: 6 +Case #4: 12 +Case #5: 33 +Case #6: 80 +Case #7: 56 +Case #8: 1000000000 +Case #9: 518198 +Case #10: 508315 +Case #11: 913948 +Case #12: 4318266 +Case #13: 4032734 +Case #14: 400462963 +Case #15: 999998019 +Case #16: 99991721 +Case #17: 99970548 +Case #18: 99653800 +Case #19: 99275145 +Case #20: 99659269 +Case #21: 1348416 +Case #22: 955943 +Case #23: 839004 +Case #24: 1711799 +Case #25: 867071 +Case #26: 1342682 +Case #27: 935396 +Case #28: 3420797 +Case #29: 73144 +Case #30: 551313 +Case #31: 841901 +Case #32: 16602 +Case #33: 545906 +Case #34: 961455 +Case #35: 308482 +Case #36: 2533 +Case #37: 351198 +Case #38: 311271 +Case #39: 851 +Case #40: 674057 +Case #41: 1724 +Case #42: 957023 +Case #43: 1050323 +Case #44: 78 +Case #45: 221298 +Case #46: 825461 +Case #47: 1423 +Case #48: 769215 +Case #49: 19836 +Case #50: 829873 +Case #51: 528866 +Case #52: 270935 +Case #53: 529324 +Case #54: 856304 +Case #55: 150966 +Case #56: 479767 +Case #57: 595329 +Case #58: 680899 +Case #59: 68255 +Case #60: 546188 +Case #61: 29151 +Case #62: 106503 +Case #63: 793385 +Case #64: 143682 +Case #65: 131026 +Case #66: 598583 +Case #67: 726888 +Case #68: 236731 +Case #69: 303583 +Case #70: 156 +Case #71: 4009 +Case #72: 494876 +Case #73: 80004 +Case #74: 5310 +Case #75: 866242 +Case #76: 621008 +Case #77: 584837 +Case #78: 790681 +Case #79: 184043 +Case #80: 5765 +Case #81: 700462 +Case #82: 964455 +Case #83: 2608 +Case #84: 371202 +Case #85: 460887 +Case #86: 341482 +Case #87: 970 +Case #88: 41473 +Case #89: 295586 +Case #90: 599224 +Case #91: 930371 +Case #92: 744812 +Case #93: 989968 +Case #94: 977190 +Case #95: 296670 +Case #96: 788102 +Case #97: 908196 +Case #98: 132070 +Case #99: 13877 +Case #100: 337998 +Case #101: 752834 +Case #102: 727583 +Case #103: 231238 +Case #104: 720664 +Case #105: 326488 +Case #106: 408477 +Case #107: 962301 +Case #108: 848828 +Case #109: 652941 +Case #110: 668060 +Case #111: 103993 +Case #112: 324617 +Case #113: 15070 diff --git a/2020/quals/timber_sol.md b/2020/quals/timber_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..c10e221c0281822edc5c56f9ab1ffb25340bbfe9 --- /dev/null +++ b/2020/quals/timber_sol.md @@ -0,0 +1,11 @@ +We can observe that any combined timber interval must consist of 0 or more trees falling to the right followed by 0 or more trees falling to the left (with these two sequences of trees meeting at a common point in between). + +Let \(R_p\) be the maximum length of a combined timber interval whose right endpoint is \(p\) and which consists only of trees cut down to the right. Let \(L_p\) similarly be the maximum length of a combined timber interval whose left endpoint is \(p\) and which consists only of trees cut down to the left. The answer (the maximum length of any combined timber interval) is then the maximum of \(( R_p + L_p )\) over all possible choices of \(p\). + +What remains is computing these \(R\) and \(L\) values. We'll consider only the \(R\) values for now, with the \(L\) values computed in a very similar fashion (imagine that the trees are reversed on the number line, such that the \(i\)th tree is at position \(-P_i\) rather than \(P_i\)). + +We'll use a dynamic programming approach, initially assuming that \(R_p = 0\) for each \(p\) and then increasing the \(R\) values as much as possible until we've considered all possible ways of increasing them. We'll first sort the trees in increasing order of position, and then iterate over them in that order. For the \(i\)th tree, a sequence of right-falling trees ending at position \(P_i\) may have that tree added onto it, meaning that we must have \(R_{P_i+H_i} \geq R_{P_i} + H_i\), and should therefore set \(R_{P_i+H_i} := max ( R_{P_i+H_i}, R_{P_i} + H_i )\). + +Note that, though \(R_p\) is defined and possibly relevant for any integer \(p\) in [\(-10^9, 10^9\)], at most \(N\) entries are non-zero (at most one for each tree). Therefore, in order to keep memory usage manageable, we should store a sparse mapping of \(p \rightarrow R_p\), for example using a hash table (such as a C++ `unordered_map` or Python `dict`). This can also help with efficient computation of \(max_p ( R_p + L_p )\), as we should only consider positions \(p\) for which either \(R_p\) or \(L_p\) is non-zero. + +The time complexity of this algorithm is \(O(N \log(N))\), resulting from sorting the trees. diff --git a/2020/quals/travel_restrictions.cpp b/2020/quals/travel_restrictions.cpp new file mode 100644 index 0000000000000000000000000000000000000000..edb6fbfe76fe138e6d40eec8821af23dad8dae1f --- /dev/null +++ b/2020/quals/travel_restrictions.cpp @@ -0,0 +1,135 @@ +// Travel Restrictions +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) { + char c, r = 0, n = 0; + x = 0; + for (;;) { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 55 + +int N; +char I[LIM], O[LIM]; +bool P[LIM][LIM]; + +void ProcessCase() +{ + // input + Read(N); + scanf("%s%s", I, O); + // simulate rightward trips + int i, j; + Fill(P, 0); + Fox(i, N) + { + j = i; + for (;;) + { + P[i][j] = 1; + if (j + 1 == N || O[j] == 'N' || I[j + 1] == 'N') + break; + j++; + } + } + // simulate leftward trips + Fox(i, N) + { + j = i; + for (;;) + { + P[i][j] = 1; + if (j - 1 < 0 || O[j] == 'N' || I[j - 1] == 'N') + break; + j--; + } + } + // output + Fox(i, N) + { + Fox(j, N) + printf("%c", P[i][j] ? 'Y' : 'N'); + printf("\n"); + } +} + +int main() { + int T, t; + Read(T); + Fox1(t, T) + { + printf("Case #%d:\n", t); + ProcessCase(); + } + return(0); +} \ No newline at end of file diff --git a/2020/quals/travel_restrictions.in b/2020/quals/travel_restrictions.in new file mode 100644 index 0000000000000000000000000000000000000000..d6f2c4dc14550b14cccea000fb4cd8d0d73e5a1c --- /dev/null +++ b/2020/quals/travel_restrictions.in @@ -0,0 +1,331 @@ +110 +10 +YYYYYYYYYY +YYYYYYYYYY +10 +NNNNNNNNNN +YYYYYYYYYY +10 +YYYYYYYYYY +NNNNNNNNNN +10 +NNNNNNNNNN +NNNNNNNNNN +10 +YNYNYNYNYN +YNYNYNYNYN +10 +YNYNYNYNYN +NYNYNYNYNY +10 +NYNYNYNYNY +YNYNYNYNYN +50 +YYNYNYYYYNYNNNYNYNYNNYYYNYYYYYYNYYYYNNYYYYYYNYYYYY +YYNYYYYYNYYYYNNYYNYNYYYYYYYNYYNNYNYNNYYYYNYYYYNYYY +49 +YYYNYYNYNYYYYYYYYYYYYYNYYYYYYYYYYYYYYYYYYYYNYNYYY +YYYYYYYYYYYYYYYYYYYYYYNYYYYYNYYYYNYYNYYYYNNYYYYYY +48 +YYYYYYYYYYYYYYYYNNYNYYYYYNYYYYYYYYNYYYYYYYYYYYYY +YYYNYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYNYYYYYYYYYY +47 +YYNNYYYYYYYNYYYYYYYYYYYYYYYYYYNYYYYYYYYYYYYYYYY +YYYYYYYYYYYYYYNYYYYYYNYYYYYYYYYYYNYYYYYNYNYYYNY +46 +YYNYNYYNYNYYNYYNYYNYNYNNNNNYNNNNYYNYNYYYYNYNNN +NYYNNYNNYYNNYNYNNNNNYNNNNNNNYYNNNYNNYNNNNYYNYN +19 +YYNYNNYYYYNYYYNYYNN +YYYYYNYNNYYYYNNYYNY +7 +NNYYYNY +NNYYNYY +11 +YYNNYNNYYNY +NYYYNNYYNYN +21 +NNNYNYNYNNNNNYYYNYNYN +NYYYNYNYYNNYNYYNNNNNY +12 +NNNNNNYYYNNN +NYYYYYNYNYYY +6 +NYYYNY +YYYYYY +10 +YNNNYNYYNN +NYNNYNYYNN +17 +YYYYYYYYYNYYYYYYY +YYYYYYYYYYYYYNYYY +12 +YYYYYYYYYYYY +NYYYYYYYNYYN +9 +YYYYYYYYY +YYYYYYYYY +17 +YYNYYNYYNYNNYYYYY +YNYYYYYYYYYYNNYYY +9 +NYYYNYYYY +YYYYYYYYY +11 +YYYYYYYNNYY +NNYYYYYNNYY +17 +NYYNYYYYNNNYNNYNY +YNYYNYNYYNYNNYYYN +13 +YYNNNYYNNYNYY +NNYYYYYYNNYNN +4 +YNYN +NYNY +16 +YYYYYYYYYYYYYYYY +YYYYYYNNYYYYYYYY +13 +YYYYNYYYYYNYY +NNNNYNNNYYNNN +18 +YYYYYYYYYYYYNNNYNY +YNYYYYNYNYYYYYNYYY +8 +NNYYYYYN +YYYYYYYY +12 +NYYYNYYYYYNN +YYYYYYNNNNYN +7 +YNYYNYN +NYYYNNN +16 +YNYYYYYYNYYYNYYY +NYNYYYYYYYYYNYYY +17 +YYYYYYYYYYYYYYYYY +YYYYYYYYYYYYYYYYY +9 +NYNNNYNYN +NNNNYNNNN +19 +NYYYYYNYNYYYYNYYYYY +YYYYYNNYYYYYYYYYYNY +9 +YYYYYYNNN +YNNYNYYYN +11 +YYYYYYYYYYY +YYYYYYYYYYY +11 +YNYNYYYYNYY +YYYYYYNYNNY +19 +YNYNYYNNNNYNNNNYNNN +NYNYYYYNYNNYNNNNNYY +15 +YYNYYYYNYYYYNYY +NYYYNYYNNYNYNYY +7 +YNYYYYY +NYYYYYY +5 +YNYYY +NNNNY +9 +NNYYYNNYN +YYYYYYYYN +12 +YYNNYYYYYNYY +NYYYYNNYYNYN +11 +YNYYYYYYYNY +NNYNNNNYYYN +16 +NNNNNNNYNYNNYNYN +YNNNYNYYYYYNNYNN +11 +YYYYYYYYYYY +YYYYYYYYYYY +15 +NYYYYNNYYYYNYYN +NNYNYNYYYNYYYNY +19 +YYYYYYYYYYYYYYYYYYY +YYYYYYYYYYYYYYYYYYY +15 +YNYNNYNNNYNNNNN +NNNYYYNYYNNYYNN +13 +YYNNNNYYYYYYN +NNNYNNYYNYNNN +14 +YYYYYYYYYYYYYY +YYYYYYYYYYYYYY +15 +YYYYYYYYYYYYYNY +YYNYYYYYYYYYYYY +21 +YYNYYYYYYYYYYYYYNYYYN +YYYYYNYYYYYYYYYYYNYYY +12 +YYNYYYYYYYYY +YYYYYYYNYYYY +5 +YYYYY +YYYYY +13 +YYYYYYYYYYYYY +YYYNYYYNNYNYY +17 +YYYYNYYYYYYNYYYYY +YYYYYYYYYYYYYYYYY +7 +YYYYYYY +YYYYYYY +17 +YYYYYYYYYYNYYNYNY +YYNYNYNYNYYNYYYNY +12 +YYNYYYYNYYYN +YYYNYNYYNYYY +5 +NNYNN +NNNNY +9 +NYYYYNNYN +YYYYYNNNY +14 +NYYNYNYNNYNYYY +YYYYYYYNYYNYYY +4 +NYYN +NNYY +10 +YYNYYYYYYY +YYYNYYYYYY +8 +YNYNNYNN +YNNNNNYY +18 +YYYNYYYYYYYYYYYYYY +YYYYYYYYYYYYNYYYYY +13 +NYYYYYYYYYYYY +NYYYYYYYYYNYY +20 +YYYNNYYYYYYYYYNNYYNN +NNYNYYNYYYYYYYYNYYNN +3 +YYY +YYY +14 +NNNYYNNYNNYNNN +NYNYNYYYYNNNNN +10 +NNNYYYYNYN +YNYYYNNYYN +10 +NNYNYNNYYY +NNYYNNYNNY +4 +NNNN +YYYY +6 +YYYNYY +YYYNYY +21 +YYYNYNYYYNYYYNYYYYYYY +YNYYYYYYYYYYYYYYNYNYY +14 +NNNYNNNYNYNYYN +NNNNNYYYYNNYNY +21 +YNYNNYYNNYYYYYYNYNNYN +YYYYYYYYYNNNNNYNYYNNY +21 +YNYYNNYYYYYYYNYNYYNYY +YYYNYYNYNNYNYYNYYYNYN +6 +YYYYYY +YYYNNY +6 +NYYYNY +YNYNNN +6 +YYNNYN +NNYYNN +6 +NYYYYY +YYYYNY +14 +YYYYNNYNYYYYNY +YYYNYYYNYYYYYY +18 +YNYYYYYYYYYNYYYNYN +YYNYYYYYYYYYNYYNYY +7 +NYYYYYY +NYYYYYN +15 +YNNYNNYNNNNYNNY +YNNNYNYNNNYNNNN +10 +NNNYNNYNNN +NNYYNNNNYN +5 +YNNNN +NNYNY +7 +YYNNNYN +NNNNNNY +6 +YYYYYY +YNYYYY +12 +NYNYYNNYNYYY +YNYNNNYNYNNN +10 +YNYYYYYYYY +YNYYYNYYYY +15 +NNNYNNNNYNYNNNN +NYNNYNNNNNNNNNY +14 +YYNYNYNYYYNNYN +YYNYNNNNNNYNYY +20 +NYYYYNYYYYYNYYYYYYYY +NYYYYYYYYYYYYYYYYYYY +12 +YYYNNYNYYYYY +YYNYYNNYYNNN +14 +YNYYYNYYYYYYNY +YNYYYNYNNNYNYY +2 +YN +YN +13 +YYNYYNYYYYYYN +NYNNYYYYYYYYY +13 +YNYYYYYYYYYYY +YYYYYYYYYYYYY +5 +YYYYY +NYNYY +9 +YYYYYYYYY +YYYYYYYYY +20 +YYNYNYNYNNYYYYYYNYNY +NYYYNYNYYNNYNYYNNYYY +50 +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +50 +YNYYYYNYNYYYNYYYYYYYYYNYYYYYYYYYYYYYYYYYYYYNYNYYYY +YYYYYYYYYYYYYYNYYYYYYYNYYYYYNYYYYNYYNYYYYNNYYYYYYY diff --git a/2020/quals/travel_restrictions.md b/2020/quals/travel_restrictions.md new file mode 100644 index 0000000000000000000000000000000000000000..a92ace356f4f717a25c0ff6d6c0be1e85a15a88a --- /dev/null +++ b/2020/quals/travel_restrictions.md @@ -0,0 +1,37 @@ +2020 has been a tough year for travelers. Air travel is especially problematic as passengers need to spend long periods in security lines, waiting areas, and crowded cabins where social distancing is difficult to maintain. + +To minimize the spread of COVID-19, each airline has decided to reorganize all of their flight routes in a linear fashion. The airlines are hoping that by not making any one country a "hub", the spread of the virus will be significantly limited. + +An airline's flights normally service \(N\) countries, running in various directions. Amidst the pandemic, each airline has carefully arranged these \(N\) countries in a sequence with each assigned a number from \(1\) to \(N\). Flights are limited to run only between pairs of countries that are adjacent in this sequence, with service in both directions. No other flights are available. That is, flights are available from country \(i\) to country \(j\) if and only if \(|i - j| = 1\). + +To make things more complicated, some countries have issued their own restrictions on incoming and outgoing travel. These restrictions are indicated by the characters \(I_{1..N}\) and \(O_{1..N}\), each of which is either *"N"* or *"Y"*: + + - If \(I_i\) = *"N"*, then incoming flights to country \(i\) from any other country are disallowed. Otherwise, if \(I_i\) = *"Y"*, they may be allowed. + - If \(O_i\) = *"N"*, then outgoing flights from country \(i\) to any other country are disallowed. Otherwise, if \(O_i\) = *"Y"*, they may be allowed. + +If a flight between adjacent countries is disallowed by the restrictions of neither its departure country nor its arrival country, then it's allowed. + +As a consulting data scientist in the airline industry, your job is to determine which trips between the various countries are possible. Let \(P_{i,j}\) = *"Y"* if it's possible to travel from country \(i\) to country \(j\) via a sequence of 0 or more flights (which may pass through other countries along the way), and \(P_{i,j}\) = *"N"* otherwise. Note that \(P_{i,i}\) is always *"Y"*. Output this \(N*N\) matrix of characters. + +# Input + +Input begins with an integer \(T\), the number of airlines. For each airline, there are three lines. The first line contains the integer \(N\). The second line contains the length-\(N\) string \(I_{1..N}\). The third line contains the length-\(N\) string \(O_{1..N}\). + +# Output + +For the \(i\)th airline, output a line containing *"Case #i:"* followed by \(N\) more lines, the \(i\)th of which contains the length-\(N\) string \(P_{i,1..N}\). + +# Constraints + +\(1 \le T \le 100\) +\(2 \le N \le 50\) + +# Explanation of Sample + +In the first case, there are two countries with no restrictions. Therefore, trips between all pairs of countries are possible. + +In the second case, there are two countries, and traveling into country 1 is restricted. Since country 2 is the only country adjacent to country 1, the only impossible trip is from country 2 to country 1. + +In the third case, there are two countries, both of which restrict inbound travel. + +In the fourth case, one may not enter countries 2 or 3, nor exit country 4. diff --git a/2020/quals/travel_restrictions.out b/2020/quals/travel_restrictions.out new file mode 100644 index 0000000000000000000000000000000000000000..2f366cfe85b3ee98736f709d98d0eb97667d18c4 --- /dev/null +++ b/2020/quals/travel_restrictions.out @@ -0,0 +1,1671 @@ +Case #1: +YYYYYYYYYY +YYYYYYYYYY +YYYYYYYYYY +YYYYYYYYYY +YYYYYYYYYY +YYYYYYYYYY +YYYYYYYYYY +YYYYYYYYYY +YYYYYYYYYY +YYYYYYYYYY +Case #2: +YNNNNNNNNN +NYNNNNNNNN +NNYNNNNNNN +NNNYNNNNNN +NNNNYNNNNN +NNNNNYNNNN +NNNNNNYNNN +NNNNNNNYNN +NNNNNNNNYN +NNNNNNNNNY +Case #3: +YNNNNNNNNN +NYNNNNNNNN +NNYNNNNNNN +NNNYNNNNNN +NNNNYNNNNN +NNNNNYNNNN +NNNNNNYNNN +NNNNNNNYNN +NNNNNNNNYN +NNNNNNNNNY +Case #4: +YNNNNNNNNN +NYNNNNNNNN +NNYNNNNNNN +NNNYNNNNNN +NNNNYNNNNN +NNNNNYNNNN +NNNNNNYNNN +NNNNNNNYNN +NNNNNNNNYN +NNNNNNNNNY +Case #5: +YNNNNNNNNN +NYNNNNNNNN +NNYNNNNNNN +NNNYNNNNNN +NNNNYNNNNN +NNNNNYNNNN +NNNNNNYNNN +NNNNNNNYNN +NNNNNNNNYN +NNNNNNNNNY +Case #6: +YNNNNNNNNN +YYYNNNNNNN +NNYNNNNNNN +NNYYYNNNNN +NNNNYNNNNN +NNNNYYYNNN +NNNNNNYNNN +NNNNNNYYYN +NNNNNNNNYN +NNNNNNNNYY +Case #7: +YYNNNNNNNN +NYNNNNNNNN +NYYYNNNNNN +NNNYNNNNNN +NNNYYYNNNN +NNNNNYNNNN +NNNNNYYYNN +NNNNNNNYNN +NNNNNNNYYY +NNNNNNNNNY +Case #8: +YYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +YYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNYYYYNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNYYYNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNYYYNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNYYYNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNYYYYYYYNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNYYYNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNYYYNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYY +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYY +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYY +Case #9: +YYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +YYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +YYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +YYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNYYYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYY +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYY +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYY +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYY +Case #10: +YYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +YYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +YYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNYYYYYYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN 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+NNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNY +Case #110: +YNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +YYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNYYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNYYYYYYYYNNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYYNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYNNNNN +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYYYY +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYY +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYY +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYY +NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNYYYY diff --git a/2020/quals/travel_restrictions_sol.md b/2020/quals/travel_restrictions_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..aa1a2c23761561c96817aaf8231030c83b1d2a5b --- /dev/null +++ b/2020/quals/travel_restrictions_sol.md @@ -0,0 +1,5 @@ +For each starting country \(s\), the set of destination countries which it's possible to reach must form some consecutive interval \(A_s..B_s\), such that \(A_s \leq s \leq B_s\). + +We can begin by assuming that \(B_s = s\), and then repeatedly increase \(B_s\) until it has reached either country \(N\) or the first country after \(s\) which is unreachable from it. As long as \(B_s < N\), it should be incremented if flights are allowed from country \(B_s\) to country \(B_s+1\), which is the case if and only if \(O_{B_s} = I_{B_s+1} =\) *"Y"*. + +The above process can be performed for each starting country \(s\), and repeated similarly to compute each \(A_s\) value. \(A_{1..N}\) and \(B_{1..N}\) may then be used to populate the required \(P\) matrix. Each part of this solution takes \(O(N^2)\) time. diff --git a/2020/round1/294725001615446.jpg b/2020/round1/294725001615446.jpg new file mode 100644 index 0000000000000000000000000000000000000000..dbe1b2f413e0c50e1493151e344e465308b086bc --- /dev/null +++ b/2020/round1/294725001615446.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:56011525b8b96d5fbe75a19e083b0204506a91b6955e0ac1ca4fff6f0d885d57 +size 6414 diff --git a/2020/round1/729459790952901.jpg b/2020/round1/729459790952901.jpg new file mode 100644 index 0000000000000000000000000000000000000000..d8b8ce28e742afd310611e32c7278be0a26d377c --- /dev/null +++ b/2020/round1/729459790952901.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:db17b13035ae318d792306c079f8605f125f76c287abadf4d061fc6439ac6bfa +size 5999 diff --git a/2020/round1/dislodging_logs.cpp b/2020/round1/dislodging_logs.cpp new file mode 100644 index 0000000000000000000000000000000000000000..05311e7e66b21a98c2946de131d26ff9cf5d13fa --- /dev/null +++ b/2020/round1/dislodging_logs.cpp @@ -0,0 +1,153 @@ +// Dislodging Logs +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 1000001 + +int N[2], K, S; +int P[2][LIM]; + +void ReadSeq(int* V, int N, int K) +{ + int i, A, B, C, D; + Fox(i, K) + Read(V[i]); + Read(A), Read(B), Read(C), Read(D); + FoxI(i, K, N - 1) + V[i] = ((LL)A * V[i - 2] + (LL)B * V[i - 1] + C) % D + 1; +} + +bool IsValid(LL m) +{ + int i, j = 0; + // iterate over log drivers from left to right + Fox(i, N[0]) + { + // greedily allocate as many clusters as possible to this log driver + int k = j; + while (k < N[1]) + { + LL d = min(Abs(P[0][i] - P[1][j]), Abs(P[0][i] - P[1][k])) + (P[1][k] - P[1][j]); + if (d > m) + break; + k++; + } + // all clusters covered? + if (k == N[1]) + return(1); + j = k; + } + return(0); +} + +LL ProcessCase() +{ + int i; + // input, and sort by position + Fox(i, 2) + Read(N[i]); + Read(K), Read(S); + Fox(i, 2) + { + ReadSeq(P[i], N[i], K); + sort(P[i], P[i] + N[i]); + } + // binary search for min. valid # of seconds + LL r1 = 1, r2 = INF; + while (r2 > r1) + { + LL m = (r1 + r2) >> 1; + if (IsValid(m)) + r2 = m; + else + r1 = m + 1; + } + return(r1); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %lld\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/round1/dislodging_logs.in b/2020/round1/dislodging_logs.in new file mode 100644 index 0000000000000000000000000000000000000000..ed940133fe155bcb0d30b96d3941aac1c2f2fea0 --- /dev/null +++ b/2020/round1/dislodging_logs.in @@ -0,0 +1,546 @@ +109 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a/2020/round1/dislodging_logs.md b/2020/round1/dislodging_logs.md new file mode 100644 index 0000000000000000000000000000000000000000..a422915714b98b428872524182d3e69c877fc4e1 --- /dev/null +++ b/2020/round1/dislodging_logs.md @@ -0,0 +1,53 @@ +The ingenuity of the great Canadian log drive is the use of natural rivers to transport logs. But as every Canadian knows, not every body of water is so free-flowing, especially in winter. In slow-moving sections of a river, it's common for the surface of the water to freeze overnight, trapping the logs. The log drivers can break up the ice with their long-handled picks, but we'll give them a helping hand anyway. [With dynamite](https://www.youtube.com/watch?v=7AheBqBk0iQ&feature=youtu.be&t=338). + +Paul is in charge of coordinating the placement of explosive charges. This is the best dynamite-related job in the log drive, not because it's the safest (which it is), but because you get to light the fuse at the end. We can model the relevant section of the river as a number line with \(N\) log drivers and \(M\) clusters of frozen logs. The \(i\)th log driver is at position \(P_i\) (measured in metres), and the \(i\)th cluster of logs is at position \(Q_i\). All \(N+M\) of these initial positions are distinct. + +Every second, each log driver can move one metre to either the left or right, or remain stationary. Multiple log drivers are permitted to occupy the same position. When a log driver occupies the same position as a cluster of frozen logs, they can spend \(S\) seconds to set an explosive charge, and can continue moving after that. **Due to an error, please note that \(S\) is always equal to \(0\).** + +As soon as an explosive charge is in place at every cluster, Paul will begin the detonation process by lighting the universal fuse (don't worry, the fuse is long enough to allow the other log drivers to clear out in time). If Paul commands the log drivers optimally, what's the minimum amount of time after which he can light the fuse? + +In order to reduce the size of the input, the positions of the logs and log drivers will not all be provided explicitly. Instead, you'll be given the first \(K\) values \(P_{1..K}\) and \(Q_{1..K}\), as well as the two quadruples of constants \((A_P, B_P, C_P, D_P)\) and \((A_Q, B_Q, C_Q, D_Q)\), and must then compute \(P_{(K+1)..N}\) and \(Q_{(K+1)..M}\) as follows: + +\(P_i = ((A_P * P_{i-2} + B_P * P_{i-1} + C_P) \text{ modulo } D_P) + 1\) for \(i \gt K\) +\(Q_i = ((A_Q * Q_{i-2} + B_Q * Q_{i-1} + C_Q) \text{ modulo } D_Q) + 1\) for \(i \gt K\) + +# Input + +Input begins with an integer \(T\), the number of frozen river sections. For each section, there are 5 lines. + +The first line contains the 4 space-separated integers \(N\), \(M\), \(K\), and \(S\). + +The second line contains the \(K\) space-separated integers \(P_{1..K}\). + +The third line contains the 4 space-separated integers \(A_P\), \(B_P\), \(C_P\), and \(D_P\). + +The fourth line contains the \(K\) space-separated integers \(Q_{1..K}\). + +The fifth line contains the 4 space-separated integers \(A_Q\), \(B_Q\), \(C_Q\), and \(D_Q\). + +# Output + +For the \(i\)th river section, print a line containing *"Case #i: "* followed by a single integer, the minimum amount of time (in seconds) before the explosive charges can all be placed and Paul can light the fuse. + +# Constraints + +\(1 \le T \le 85\) +\(2 \le N, M \le 1,000,000\) +\(2 \le K \le \text{min}(N, M)\) +\(S = 0\) +\(0 \le A_P, B_P, C_P, A_Q, B_Q, C_Q \le 1,000,000,000\) +\(1 \le D_P, D_Q \le 500,000,000\) +\(1 \le P_i \le D_P\) +\(1 \le Q_i \le D_Q\) + +The sum of \(N+M\) across all test cases is at most 10,000,000. + +# Explanation of Sample + +For the first river section, \(P = [10, 20]\) and \(Q = [13, 16]\). It's optimal for the first log driver to walk to the first log cluster (at position 13) in 3 seconds and set an explosive charge there, while the second log driver walks to the second cluster in 4 seconds and does the same. After 4 seconds, both explosive charges will have been placed, at which point Paul can light the fuse. + +For the second river section, it's optimal for the first log driver to take the day off while the second walks to the second log cluster, sets an explosive charge there, and then walks to the first cluster and does the same. This process will take a total of 2 + 1 = 3 seconds. + +For the fourth river section: +\(P = [83, 16, 59, 131, 222, 371, 229, 214]\) +\(Q = [8, 62, 92, 202, 70, 150, 250]\) diff --git a/2020/round1/dislodging_logs.out b/2020/round1/dislodging_logs.out new file mode 100644 index 0000000000000000000000000000000000000000..489ab25b6dd9efd0269ecf99726d15434d316b7f --- /dev/null +++ b/2020/round1/dislodging_logs.out @@ -0,0 +1,109 @@ +Case #1: 4 +Case #2: 3 +Case #3: 15 +Case #4: 21 +Case #5: 229 +Case #6: 39 +Case #7: 13013924 +Case #8: 8686875 +Case #9: 100845145 +Case #10: 2 +Case #11: 89999916 +Case #12: 100299951 +Case #13: 357191123 +Case #14: 9549757 +Case #15: 249999999 +Case #16: 242010158 +Case #17: 35175200 +Case #18: 207917829 +Case #19: 71747265 +Case #20: 232926833 +Case #21: 124807267 +Case #22: 18691514 +Case #23: 16730174 +Case #24: 280828443 +Case #25: 38934152 +Case #26: 307872561 +Case #27: 12884389 +Case #28: 230863848 +Case #29: 23205893 +Case #30: 25116840 +Case #31: 182051708 +Case #32: 77646393 +Case #33: 158483945 +Case #34: 31981766 +Case #35: 158830353 +Case #36: 146606581 +Case #37: 74034802 +Case #38: 236783733 +Case #39: 156573152 +Case #40: 293932214 +Case #41: 225041057 +Case #42: 326075158 +Case #43: 207167546 +Case #44: 163305844 +Case #45: 273759394 +Case #46: 197131763 +Case #47: 30065101 +Case #48: 146110428 +Case #49: 54491591 +Case #50: 286931736 +Case #51: 24144009 +Case #52: 44303533 +Case #53: 20764974 +Case #54: 167137349 +Case #55: 161696875 +Case #56: 225877079 +Case #57: 8076425 +Case #58: 54458980 +Case #59: 68070742 +Case #60: 106005205 +Case #61: 33176981 +Case #62: 26620315 +Case #63: 382438100 +Case #64: 15369140 +Case #65: 330884145 +Case #66: 353282412 +Case #67: 218017830 +Case #68: 110106357 +Case #69: 195354283 +Case #70: 56834934 +Case #71: 23568056 +Case #72: 65440662 +Case #73: 18591467 +Case #74: 182348575 +Case #75: 24049866 +Case #76: 339313049 +Case #77: 22871972 +Case #78: 45393801 +Case #79: 299034115 +Case #80: 163695201 +Case #81: 216088366 +Case #82: 41469954 +Case #83: 219751666 +Case #84: 352332037 +Case #85: 74818449 +Case #86: 369170658 +Case #87: 72608895 +Case #88: 317743773 +Case #89: 359108677 +Case #90: 151192776 +Case #91: 72058321 +Case #92: 22926992 +Case #93: 241862339 +Case #94: 25868778 +Case #95: 290883404 +Case #96: 14146612 +Case #97: 15450674 +Case #98: 183571631 +Case #99: 51716680 +Case #100: 111175676 +Case #101: 37192026 +Case #102: 325763043 +Case #103: 197223933 +Case #104: 26529674 +Case #105: 39688457 +Case #106: 33905127 +Case #107: 17714894 +Case #108: 218030830 +Case #109: 262442354 diff --git a/2020/round1/dislodging_logs_sol.md b/2020/round1/dislodging_logs_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..c9ca3d9a1ee0a141e5f0bfb59a7c87ed7f570166 --- /dev/null +++ b/2020/round1/dislodging_logs_sol.md @@ -0,0 +1,13 @@ +Let \(F(t)\) equal \(true\) if it's possible to light the fuse within at most \(t\) seconds, and equal to \(false\) otherwise. If \(F(t_1) = true\), this implies that \(F(t_2) = true\) for every \(t_2 \ge t_1\). This means that it's possible to binary search for the smallest value of \(t\) such that \(F(t) = true\), which will be our answer. This value must be positive and no greater than \(D_P + D_Q\). What remains is efficient evaluation of \(F(t)\). + +Let \(E_i\) be the set of log cluster positions which the \(i\)th log driver will set explosive charges at. If \(E_i\) is non-empty, we can observe that the number of seconds the \(i\)th log driver requires is equal to \(\text{min}(|\text{min}(E_i) - P_i|, |\text{max}(E_i) - P_i|)\) \(+\) \((\text{max}(E_i) - \text{min}(E_i))\), where \(\text{min}(E_i)\) and \(\text{max}(E_i)\) are the smallest and largest positions in \(E_i\), respectively. + +This observation may be turned into a key insight — if we consider the log drivers and the log clusters each sorted in increasing order of position, then it's optimal for \(E_{1..N}\) to correspond to a partioning of the log clusters into a sequence of contiguous subarrays handled by one log driver each, in the same relative order. Interleaving two log drivers' \(E\) sets (such that either of them no longer consists of a subarray of log clusters) can only increase the time required, as can swapping two \(E\) sets (such that an earlier log driver sets explosive charges at a later subarray of log clusters). + +Based on this insight, we can iterate over the log drivers in increasing order of position, while maintaining a non-decreasing pointer into the sorted list of log clusters and greedily allocating as many of the next log clusters as possible to the current log driver (without them requiring more than \(t\) seconds to handle those log clusters). By the time all \(N\) log drivers have been considered, we must have \(F(t) = true\) if and only if all \(M\) log clusters have been allocated to them. + +Evaluating \(F(t)\) takes just \(O(N+M)\) time, assuming that the \(P\) and \(Q\) values have already been sorted. As such, the overall time complexity of this solution comes out to \(O(N log(N) + M log(M) + (N+M) log(D_P + D_Q))\). + +Though we initially believed that the above solution extends to the case in which \(S > 0\), it does not, as it is then no longer necessarily the case that the partitions of logs clusters should be assigned to log drivers in the same relative order. For example, consider the counterexample (pointed out by competitor Scott Wu) in which there are 2 log drivers at positions \(20\) and \(25\), 3 log clusters at positions \(5\), \(10\), and \(11\), and \(S = 10\). In this case, the first log cluster should be handled by the second log driver to achieve the optimal answer of 30. + +Because this error unfortunately only came to light during the round, we altered the constraints at that point such that \(S\) is always equal to \(0\) rather than being positive. Since everybody who received an Accepted verdict before that point had written a solution trivially applicable to \(S = 0\) and consistent with our intended solution, we kept those Accepted verdicts, while rejudging all submissions which had previously received Wrong Answer verdicts. diff --git a/2020/round1/perimetric_ch1.cpp b/2020/round1/perimetric_ch1.cpp new file mode 100644 index 0000000000000000000000000000000000000000..66a636d27380540fca28983e7d87d0ccca954c5d --- /dev/null +++ b/2020/round1/perimetric_ch1.cpp @@ -0,0 +1,146 @@ +// Perimetric - Chapter 1 +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 1000001 +#define MOD 1000000007 + +int W; +LL cur, ans; +int L[LIM], H[LIM]; +PR Q[LIM]; +int Qs, Qe; + +void ReadSeq(int* V, int N, int K) +{ + int i, A, B, C, D; + Fox(i, K) + Read(V[i]); + Read(A), Read(B), Read(C), Read(D); + FoxI(i, K, N - 1) + V[i] = ((LL)A * V[i - 2] + (LL)B * V[i - 1] + C) % D + 1; +} + +void ProcessRect(int L, int H) +{ + // remove rectangles which are no longer relevant (too far left) + while (Qe != Qs && Q[Qs].x < L) + Qs++; + if (Qe == Qs) + cur += W + H; // no overlap, so add full rectangle perimeter + else + { + cur += L + W - Q[Qe - 1].x; // add additional horizontal perimeter + cur += max(0, H - Q[Qs].y); // add additional vertical perimeter + } + // remove rectangles which are no longer relevant (shorter than this one) + while (Qe != Qs && Q[Qe - 1].y <= H) + Qe--; + // add this rectangle + Q[Qe++] = mp(L + W, H); +} + +int ProcessCase() +{ + int N, K, i; + // input + Read(N), Read(K), Read(W); + ReadSeq(L, N, K); + ReadSeq(H, N, K); + // process rectangles, while maintaining queue of relevant ones + cur = 0, ans = 1; + Qs = Qe = 0; + Fox(i, N) + { + ProcessRect(L[i], H[i]); + ans = ans * (cur % MOD) * 2 % MOD; + } + return(ans); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/round1/perimetric_ch1.in b/2020/round1/perimetric_ch1.in new file mode 100644 index 0000000000000000000000000000000000000000..49a7214a123692ea162a011e450b09ec50dfa4ab --- /dev/null +++ b/2020/round1/perimetric_ch1.in @@ -0,0 +1,541 @@ +108 +2 2 2 +1 2 +0 0 0 100 +3 3 +0 0 0 100 +2 2 2 +10 20 +0 0 0 100 +3 3 +0 0 0 100 +5 5 3 +2 4 5 9 12 +0 0 0 100 +4 3 6 3 2 +0 0 0 100 +10 3 8 +9 14 15 +0 1 3 53 +12 7 16 +5 2 1 38 +50 10 17 +4 9 10 26 28 59 97 100 105 106 +1 0 7 832 +130 12 82 487 12 30 214 104 104 527 +21 81 410 605 +200 52 18 +524 546 576 601 611 631 652 661 694 719 722 732 763 770 785 789 790 796 827 829 851 869 880 898 912 936 940 944 970 993 1009 1025 1029 1033 1048 1082 1094 1103 1129 1158 1169 1186 1203 1231 1252 1283 1287 1319 1347 1349 1383 1414 +0 1 16 41948 +24214 5498 9166 7368 340 278 651 3188 730 4665 2384 732 6097 1831 4420 9037 772 8151 515 82 4424 113 6023 427 614 431 155 286 702 1015 2451 352 604 9026 13339 24134 960 558 194 2797 4460 629 2861 644 7396 961 407 533 6628 166 559 2344 +128277793 281088604 354280740 26747 +1000 95 1 +6887 6888 6890 6892 6894 6896 6897 6899 6901 6902 6904 6906 6907 6908 6910 6912 6913 6914 6916 6917 6919 6920 6922 6923 6925 6927 6929 6931 6933 6934 6935 6937 6939 6941 6943 6944 6946 6948 6950 6952 6954 6956 6957 6959 6960 6962 6964 6965 6967 6968 6970 6971 6973 6974 6975 6977 6978 6979 6981 6983 6985 6986 6988 6990 6992 6993 6995 6997 6999 7000 7002 7004 7005 7006 7008 7009 7010 7011 7013 7014 7016 7017 7019 7020 7021 7022 7023 7025 7027 7028 7029 7031 7032 7034 7036 +0 1 1 164696450 +791377 171463 191844 683116 390069 290020 346452 227385 427109 955550 627491 55702 877581 719516 386765 783694 751441 993533 677929 930982 473502 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The solution to any chapter may help with solving the others, so please consider reading all of them.** + +After a highly successful haul from the log drive, Connie the contractor is tasked with building a number of houses in the Great White North. For each job, the client has provided a floor plan consisting of \(N\) rectangular rooms, numbered from 1 to \(N\). From a bird's-eye view, the rooms are arranged on a 2-dimensional plane, with axis-aligned walls. The southern wall of each room has y-coordinate 0. + +The \(i\)th rectangular room has southwest corner \((L_i, 0)\) and northeast corner \((L_i + W, H_i)\). **In this chapter of the problem, all \(N\) rooms have the same width \(W\), and have strictly increasing \(L\) values \((L_1 < L_2 < ... < L_N)\).** Since houses often have shared regions (such as a common living/dining area), these rooms may overlap with one another. + +Unfortunately, [log houses are quite susceptible to air leakage](https://www.energy.gov/energysaver/types-homes/energy-efficiency-log-homes). Connie knows that she must install additional insulation to keep the houses warm and energy-efficient during the harsh Canadian winters. In order to determine the amount of insulation material required, Connie will first need to gather some metrics: the perimeters around various combinations of rooms. + +Specifically, let \(P_i\) be the perimeter of the union of rooms \(1..i\). Note that any given point is considered to be within the union if and only if it's within at least one of the rooms' rectangles (including right on an edge), and that the union might not form a single connected polygon. Please help compute the product \((P_1 * P_2 * ... * P_N)\). As this product may be very large, you should compute its value modulo 1,000,000,007. + +In order to reduce the size of the input, the rooms' coordinates will not all be provided explicitly. Instead, you'll be given the first \(K\) values \(L_{1..K}\) and \(H_{1..K}\), as well as the two quadruples of constants \((A_L, B_L, C_L, D_L)\) and \((A_H, B_H, C_H, D_H)\), and must then compute \(L_{(K+1)..N}\) and \(H_{(K+1)..N}\) as follows: + +\(L_i = ((A_L * L_{i-2} + B_L * L_{i-1} + C_L) \text{ modulo } D_L) + 1\) for \(i > K\) +\(H_i = ((A_H * H_{i-2} + B_H * H_{i-1} + C_H) \text{ modulo } D_H) + 1\) for \(i > K\) + +# Input + +Input begins with an integer \(T\), the number of floor plans. For each plan, there are 5 lines. + +The first line contains the 3 space-separated integers \(N\), \(K\), and \(W\). + +The second line contains the \(K\) space-separated integers \(L_{1..K}\). + +The third line contains the 4 space-separated integers \(A_L\), \(B_L\), \(C_L\), and \(D_L\). + +The fourth line contains the \(K\) space-separated integers \(H_{1..K}\). + +The fifth line contains the 4 space-separated integers \(A_H\), \(B_H\), \(C_H\), and \(D_H\). + +# Output + +For the \(i\)th floor plan, print a line containing *"Case #i: "* followed by a single integer, the product \((P_1 * P_2 * ... * P_N)\) as defined above, modulo 1,000,000,007. + +# Constraints + +\(1 \le T \le 100\) +\(2 \le N \le 1,000,000\) +\(2 \le K \le N\) +\(1 \le W \le 20\) +\(0 \le A_L, B_L, C_L, A_H, B_H, C_H \le 1,000,000,000\) +\(1 \le D_L, D_H \le 500,000,000\) +\(1 \le L_i \le D_L\) +\(1 \le H_i \le D_H\) +\(L_1 < L_2 < ... < L_N\) + +The sum of \(N\) across all floor plans is at most 10,000,000. + +# Explanation of Sample + +For the first floor plan, \(L = [1, 2]\) and \(H = [3, 3]\). The union of just the first room's rectangle is equivalent to that rectangle, which has a width of 2 and a height of 3, and therefore a perimeter of 10. The union of both rooms' rectangles is equivalent to a single rectangle with a width of 3 and a height of 3 (with southwest corner (1, 0)), having a perimeter of 12. The answer is therefore \((10 * 12)\text{ modulo }1\text{,}000\text{,}000\text{,}007 = 120\). + +For the second floor plan, \(P = [10, 20]\). + +For the third floor plan, \(P = [14, 18, 24, 36, 42]\). + +For the fourth floor plan: +\(L = [9, 14, 15, 19, 23, 27, 31, 35, 39, 43]\) +\(H = [12, 7, 16, 31, 30, 27, 16, 17, 2, 15]\) +\(P = [40, 50, 60, 98, 106, 114, 122, 130, 138, 146]\) diff --git a/2020/round1/perimetric_ch1.out b/2020/round1/perimetric_ch1.out new file mode 100644 index 0000000000000000000000000000000000000000..45070f26ab64a51ff8ec28a7650eaf2471147e60 --- /dev/null +++ b/2020/round1/perimetric_ch1.out @@ -0,0 +1,108 @@ +Case #1: 120 +Case #2: 200 +Case #3: 9144576 +Case #4: 803986060 +Case #5: 271473330 +Case #6: 491386184 +Case #7: 298563205 +Case #8: 330339539 +Case #9: 750990988 +Case #10: 178008332 +Case #11: 109716482 +Case #12: 442862902 +Case #13: 867071613 +Case #14: 766812024 +Case #15: 58721625 +Case #16: 828709315 +Case #17: 537279442 +Case #18: 408452315 +Case #19: 554808246 +Case #20: 589809863 +Case #21: 272317923 +Case #22: 762901809 +Case #23: 793560650 +Case #24: 837011765 +Case #25: 557429595 +Case #26: 435799338 +Case #27: 570346441 +Case #28: 484594874 +Case #29: 867922559 +Case #30: 692124657 +Case #31: 967965940 +Case #32: 199567183 +Case #33: 899545163 +Case #34: 796963765 +Case #35: 242616337 +Case #36: 337452373 +Case #37: 304560211 +Case #38: 257218855 +Case #39: 604749624 +Case #40: 595985687 +Case #41: 432201331 +Case #42: 696032193 +Case #43: 392477448 +Case #44: 70594822 +Case #45: 583938261 +Case #46: 571921048 +Case #47: 334235631 +Case #48: 110527727 +Case #49: 476306795 +Case #50: 159918669 +Case #51: 934863235 +Case #52: 227069633 +Case #53: 241524091 +Case #54: 185886782 +Case #55: 470510979 +Case #56: 812726889 +Case #57: 155526718 +Case #58: 424736751 +Case #59: 844452827 +Case #60: 610240249 +Case #61: 655305021 +Case #62: 354056608 +Case #63: 713515056 +Case #64: 84867200 +Case #65: 993340372 +Case #66: 243336237 +Case #67: 961588361 +Case #68: 895259827 +Case #69: 362275837 +Case #70: 39985128 +Case #71: 856024293 +Case #72: 321459754 +Case #73: 395528589 +Case #74: 77024378 +Case #75: 132733964 +Case #76: 805611316 +Case #77: 186976437 +Case #78: 685368574 +Case #79: 640876383 +Case #80: 455650376 +Case #81: 999913095 +Case #82: 989608453 +Case #83: 786029527 +Case #84: 929299681 +Case #85: 783150695 +Case #86: 62320419 +Case #87: 141320917 +Case #88: 824272480 +Case #89: 945744989 +Case #90: 120219020 +Case #91: 321026168 +Case #92: 789649723 +Case #93: 967085124 +Case #94: 185739339 +Case #95: 162925250 +Case #96: 602657809 +Case #97: 421142722 +Case #98: 427823292 +Case #99: 284740895 +Case #100: 37872339 +Case #101: 657264215 +Case #102: 973061357 +Case #103: 232550834 +Case #104: 531859369 +Case #105: 493298602 +Case #106: 919127854 +Case #107: 935419278 +Case #108: 690958334 diff --git a/2020/round1/perimetric_ch1_sol.md b/2020/round1/perimetric_ch1_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..3165ba1cde62da428dc21d72ddac6da9139bc11e --- /dev/null +++ b/2020/round1/perimetric_ch1_sol.md @@ -0,0 +1,9 @@ +We'll process the rectangles one by one while computing \(P_{1..N}\). When processing the \(i\)th rectangle, we can't afford to compute the perimeter of the union of the first \(i\) rectangles from scratch (as this would result in an inefficient algorithm with time complexity at least \(O(N^2)\)), so we'll instead compute how the inclusion of this rectangle affects the existing rectangle union (and therefore how \(P_i\) differs from \(P_{i-1}\), assuming that \(P_0 = 0\)). + +If the \(i\)th rectangle does not overlap (inclusively) with any existing rectangles (in other words, if either \(i = 1\) or \(L_i > L_{i-1} + W\)), then its entire perimeter will be added to the union's perimeter: \(P_i = P_{i-1} + 2(W + H_i)\). + +Otherwise, let's separately consider the amount of horizontal and vertical perimeter added. The amount of additional horizontal perimeter is simply the amount of extra width that the union's perimeter spans, multiplied by two (to account for both the top and the bottom): \(2(L_i - L_{i-1})\). Meanwhile, the amount of additional vertical perimeter is dependent on the largest height \(h\) of any existing rectangle which the \(i\)th one overlaps (inclusively) with. Specifically, if \(H_i \le h\), then there is no additional vertical perimeter (as the top of the union still only comes down from \(h\) to \(0\)), while there will otherwise be \(2(H_i - h)\) additional vertical perimeter (as the top of the union goes up to a height of \(H_i\) and then back down). + +The simplest way to compute \(h\) in the above expression is to loop back over all of the rectangles overlapping with the \(i\)th one, which are at most the last \(W\) ones. This approach results in a time complexity of \(O(NW)\), which is sufficient. + +However, it can also be optimized down to just \(O(N)\) by maintaining a deque of rectangles which could possibly be relevant for this purpose in the future (that is, ones whose \(L\) values are no less than \(L_i - W\), and which have both increasing \(L\) values and decreasing \(H\) values). The rectangle with the largest height overlapping with the \(i\)th one is then known to be the earliest one in this deque. diff --git a/2020/round1/perimetric_ch2.cpp b/2020/round1/perimetric_ch2.cpp new file mode 100644 index 0000000000000000000000000000000000000000..cc241a36ae31def5cf88ae3403c62cb757c95c65 --- /dev/null +++ b/2020/round1/perimetric_ch2.cpp @@ -0,0 +1,147 @@ +// Perimetric - Chapter 2 +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 1000001 +#define MOD 1000000007 + +LL cur, ans; +int L[LIM], W[LIM], H[LIM]; +set S; + +void ReadSeq(int* V, int N, int K) +{ + int i, A, B, C, D; + Fox(i, K) + Read(V[i]); + Read(A), Read(B), Read(C), Read(D); + FoxI(i, K, N - 1) + V[i] = ((LL)A * V[i - 2] + (LL)B * V[i - 1] + C) % D + 1; +} + +void ProcessRect(int L, int R, int H) +{ + // find first x interval overlapping with this rectangle + auto I = prev(S.lower_bound(mp(L, -1))); + if (I->y < L) + I++; + Min(L, I->x); + // erase all x intervals overlapping with this rectangle + while (I->x <= R) + { + Max(R, I->y); + cur -= I->y - I->x + H; + I = S.erase(I); + } + // add x interval for this rectangle (and any intervals it overlapped) + cur += R - L + H; + S.insert(mp(L, R)); +} + +int ProcessCase() +{ + int N, K, i; + // input + Read(N), Read(K); + ReadSeq(L, N, K); + ReadSeq(W, N, K); + ReadSeq(H, N, K); + // process rectangles, while maintaining set of covered x intervals + cur = 0, ans = 1; + S.clear(); + S.insert(mp(-1, -1)); + S.insert(mp(INF + 1, INF + 1)); + Fox(i, N) + { + ProcessRect(L[i], L[i] + W[i], H[i]); + ans = (LL)ans * (cur % MOD) * 2 % MOD; + } + return(ans); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/round1/perimetric_ch2.in b/2020/round1/perimetric_ch2.in new file mode 100644 index 0000000000000000000000000000000000000000..d51175e5e02e698ad257142b2b8a41b4ba501227 --- /dev/null +++ b/2020/round1/perimetric_ch2.in @@ -0,0 +1,757 @@ +108 +2 2 +1 2 +0 0 0 100 +2 2 +0 0 0 100 +3 3 +0 0 0 100 +2 2 +10 20 +0 0 0 100 +2 2 +0 0 0 100 +3 3 +0 0 0 100 +5 5 +8 3 10 16 2 +0 0 0 100 +3 1 5 2 18 +0 0 0 100 +9 6 6 4 1 +0 0 0 100 +10 3 +14 5 14 +4 7 2 47 +6 4 13 +2 3 17 33 +29 23 22 +1 0 27 31 +50 10 +100 26 4 28 106 59 9 105 10 97 +14 40 784 4832 +130 12 82 487 12 30 214 104 104 527 +21 81 410 605 +938 903 899 899 896 857 841 837 834 834 +1 0 924 951 +200 22 +1540 1303 108 1590 1053 130 149 1065 1816 217 15 2075 1258 654 1162 1280 1346 584 771 1388 725 1479 +197759654 912778406 625448953 15461 +1133 236 896 9649 5182 3599 7175 5911 3974 1825 2501 437 5419 7655 997 4781 9394 6489 4997 1590 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40269484 40269483 40269480 40269472 40269464 40269457 40269454 40269445 40269439 40269436 40269431 40269424 40269418 40269412 40269412 40269411 40269405 40269405 40269402 40269396 40269389 40269383 40269383 40269374 40269369 40269366 40269366 40269358 40269358 40269353 40269351 40269344 40269338 40269338 40269332 40269329 40269325 40269320 40269311 40269305 40269299 40269299 40269299 40269291 40269289 40269285 40269278 40269273 40269264 40269258 40269250 40269245 40269238 40269232 40269232 40269228 40269220 40269218 40269213 40269208 40269207 40269204 40269202 40269201 40269192 40269192 40269191 40269185 40269180 40269176 40269171 40269165 40269159 40269152 40269151 40269143 40269139 40269136 40269127 40269119 40269115 40269112 40269104 40269099 40269090 40269089 40269083 40269081 40269080 40269074 +0 1 57950800 57950815 diff --git a/2020/round1/perimetric_ch2.md b/2020/round1/perimetric_ch2.md new file mode 100644 index 0000000000000000000000000000000000000000..5ee34c2fdf1747d679651242e0a1302555dfcb82 --- /dev/null +++ b/2020/round1/perimetric_ch2.md @@ -0,0 +1,65 @@ +**Note: This problem shares similarities with Chapters [1](https://www.facebook.com/codingcompetitions/hacker-cup/2020/round-1/problems/A1) and [3](https://www.facebook.com/codingcompetitions/hacker-cup/2020/round-1/problems/A3). The solution to any chapter may help with solving the others, so please consider reading all of them.** + +After a highly successful haul from the log drive, Connie the contractor is tasked with building a number of houses in the Great White North. For each job, the client has provided a floor plan consisting of \(N\) rectangular rooms, numbered from 1 to \(N\). From a bird's-eye view, the rooms are arranged on a 2-dimensional plane, with axis-aligned walls. The southern wall of each room has y-coordinate 0. + +The \(i\)th rectangular room has southwest corner \((L_i, 0)\) and northeast corner \((L_i + W_i, H_i\)). **In this chapter of the problem, the rooms have non-increasing \(H\) values \((H_1 \ge H_2 \ge ... \ge H_N)\).** Since houses often have shared regions (such as a common living/dining area), these rooms may overlap with one another. + +Unfortunately, [log houses are quite susceptible to air leakage](https://www.energy.gov/energysaver/types-homes/energy-efficiency-log-homes). Connie knows that she must install additional insulation to keep the houses warm and energy-efficient during the harsh Canadian winters. In order to determine the amount of insulation material required, Connie will first need to gather some metrics: the perimeters around various combinations of rooms. + +Specifically, let \(P_i\) be the perimeter of the union of rooms \(1..i\). Note that any given point is considered to be within the union if and only if it's within at least one of the rooms' rectangles (including right on an edge), and that the union might not form a single connected polygon. Please help compute the product \((P_1 * P_2 * ... * P_N)\). As this product may be very large, you should compute its value modulo 1,000,000,007. + +In order to reduce the size of the input, the rooms' coordinates will not all be provided explicitly. Instead, you'll be given the first \(K\) values \(L_{1..K}\), \(W_{1..K}\), and \(H_{1..K}\), as well as the three quadruples of constants \((A_L, B_L, C_L, D_L)\), \((A_W, B_W, C_W, D_W)\), and \((A_H, B_H, C_H, D_H)\), and must then compute \(L_{(K+1)..N}\), \(W_{(K+1)..N}\), and \(H_{(K+1)..N}\) as follows: + +\(L_i = ((A_L * L_{i-2} + B_L * L_{i-1} + C_L) \text{ modulo } D_L) + 1\) for \(i > K\) +\(W_i = ((A_W * W_{i-2} + B_W * W_{i-1} + C_W) \text{ modulo } D_W) + 1\) for \(i > K\) +\(H_i = ((A_H * H_{i-2} + B_H * H_{i-1} + C_H) \text{ modulo } D_H) + 1\) for \(i > K\) + +# Input + +Input begins with an integer \(T\), the number of floor plans. For each plan, there are 7 lines. + +The first line contains the 2 space-separated integers \(N\) and \(K\). + +The second line contains the \(K\) space-separated integers \(L_{1..K}\). + +The third line contains the 4 space-separated integers \(A_L\), \(B_L\), \(C_L\), and \(D_L\). + +The fourth line contains the \(K\) space-separated integers \(W_{1..K}\). + +The fifth line contains the 4 space-separated integers \(A_W\), \(B_W\), \(C_W\), and \(D_W\). + +The sixth line contains the \(K\) space-separated integers \(H_{1..K}\). + +The seventh line contains the 4 space-separated integers \(A_H\), \(B_H\), \(C_H\), and \(D_H\). + +# Output + +For the \(i\)th floor plan, print a line containing *"Case #i: "* followed by a single integer, the product \((P_1 * P_2 * ... * P_N)\) as defined above, modulo 1,000,000,007. + +# Constraints + +\(1 \le T \le 95\) +\(2 \le N \le 1,000,000\) +\(2 \le K \le N\) +\(0 \le A_L, B_L, C_L, A_W, B_W, C_W, A_H, B_H, C_H \le 1,000,000,000\) +\(1 \le D_L, D_W, D_H \le 500,000,000\) +\(1 \le L_i \le D_L\) +\(1 \le W_i \le D_W\) +\(1 \le H_i \le D_H\) +\(H_1 \ge H_2 \ge ... \ge H_N\) + +The sum of \(N\) across all floor plans is at most 10,000,000. + +# Explanation of Sample + +For the first floor plan, \(L = [1, 2]\) and \(H = [3, 3]\). The union of just the first room's rectangle is equivalent to that rectangle, which has a width of 2 and a height of 3, and therefore a perimeter of 10. The union of both rooms' rectangles is equivalent to a single rectangle with a width of 3 and a height of 3 (with southwest corner (1, 0)), having a perimeter of 12. The answer is therefore \((10 * 12)\text{ modulo }1\text{,}000\text{,}000\text{,}007 = 120\). + +For the second floor plan, \(P = [10, 20]\). + +For the third floor plan, \(P = [24, 38, 46, 58, 70]\). + +For the fourth floor plan: +\(L = [14, 5, 14, 27, 13, 14, 12, 2, 18, 43]\) +\(W = [6, 4, 13, 32, 8, 7, 22, 32, 26, 28]\) +\(H = [29, 23, 22, 20, 19, 17, 16, 14, 13, 11]\) +\(P = [70, 124, 138, 202, 204, 204, 206, 190, 190, 214]\) diff --git a/2020/round1/perimetric_ch2.out b/2020/round1/perimetric_ch2.out new file mode 100644 index 0000000000000000000000000000000000000000..7817e66188d67497678ac3ca404bffb9bae1503f --- /dev/null +++ b/2020/round1/perimetric_ch2.out @@ -0,0 +1,108 @@ +Case #1: 120 +Case #2: 200 +Case #3: 170325120 +Case #4: 81786362 +Case #5: 950017190 +Case #6: 30538611 +Case #7: 941855890 +Case #8: 126863246 +Case #9: 313131619 +Case #10: 646192635 +Case #11: 471168914 +Case #12: 399092530 +Case #13: 525192634 +Case #14: 877031788 +Case #15: 474514231 +Case #16: 217330121 +Case #17: 174120031 +Case #18: 715944550 +Case #19: 310049399 +Case #20: 900971938 +Case #21: 584442831 +Case #22: 399469835 +Case #23: 612722710 +Case #24: 533934511 +Case #25: 556159978 +Case #26: 100226849 +Case #27: 134170215 +Case #28: 467515014 +Case #29: 764061936 +Case #30: 920666309 +Case #31: 842535239 +Case #32: 786458591 +Case #33: 194240144 +Case #34: 104922004 +Case #35: 22079001 +Case #36: 399722280 +Case #37: 366908081 +Case #38: 244635847 +Case #39: 909442520 +Case #40: 597273714 +Case #41: 766095988 +Case #42: 705139041 +Case #43: 72469920 +Case #44: 936898139 +Case #45: 264413652 +Case #46: 722582554 +Case #47: 874826843 +Case #48: 694480123 +Case #49: 185346967 +Case #50: 984403906 +Case #51: 541919368 +Case #52: 623538358 +Case #53: 756914000 +Case #54: 988600035 +Case #55: 820828408 +Case #56: 83744686 +Case #57: 893903089 +Case #58: 724963691 +Case #59: 102236110 +Case #60: 563239373 +Case #61: 944427599 +Case #62: 278151733 +Case #63: 327421383 +Case #64: 89090968 +Case #65: 665024609 +Case #66: 751978467 +Case #67: 243079987 +Case #68: 232742380 +Case #69: 539793199 +Case #70: 991231342 +Case #71: 755010570 +Case #72: 757720939 +Case #73: 366773249 +Case #74: 518454023 +Case #75: 744213980 +Case #76: 804744235 +Case #77: 544448407 +Case #78: 150284162 +Case #79: 873026669 +Case #80: 141510801 +Case #81: 528859330 +Case #82: 411361470 +Case #83: 576245178 +Case #84: 212474011 +Case #85: 576241012 +Case #86: 926796293 +Case #87: 736548144 +Case #88: 590429247 +Case #89: 235971823 +Case #90: 585841131 +Case #91: 440711266 +Case #92: 35383003 +Case #93: 204994779 +Case #94: 846679613 +Case #95: 97444013 +Case #96: 565061483 +Case #97: 23305804 +Case #98: 117535895 +Case #99: 481009814 +Case #100: 229555147 +Case #101: 325902652 +Case #102: 645041361 +Case #103: 451311151 +Case #104: 450916571 +Case #105: 968446207 +Case #106: 864463399 +Case #107: 231588200 +Case #108: 16848489 diff --git a/2020/round1/perimetric_ch2_sol.md b/2020/round1/perimetric_ch2_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..07543babfc5f376f358a70be8a77d512025cead0 --- /dev/null +++ b/2020/round1/perimetric_ch2_sol.md @@ -0,0 +1,7 @@ +As in Chapter 1 of this problem, we'll process the rectangles one by one while computing \(P_{1..N}\). When processing the \(i\)th rectangle, we can't afford to compute the perimeter of the union of the first \(i\) rectangles from scratch (as this would result in an inefficient algorithm with time complexity at least \(O(N^2)\)), so we'll instead compute how the inclusion of this rectangle affects the existing rectangle union (and therefore how \(P_i\) differs from \(P_{i-1}\), assuming that \(P_0 = 0\)). + +Because each rectangle is no taller than all existing ones, the exact heights of those existing ones end up not mattering — only their x-coordinates do. As such, we'll maintain an ordered set of non-intersecting intervals of x-coordinates which the rectangle union spans. + +Let's separately consider the amount of horizontal and vertical perimeter added for the \(i\)th rectangle. The total horizontal perimeter of the rectangle union is always just the sum of the lengths in the intervals in our set, multiplied by two (to account for both the top and the bottom). Meanwhile, we can observe that the \(i\)th rectangle contributes \(2H_i\) vertical perimeter, minus \(2H_i\) for each existing interval that it overlaps (inclusively) with (as the perimeter is prevented from going all the way down to y-coordinate \(0\) and back up to \(H_i\). + +Based on the above observations, as long as we're able to update the ordered set of intervals for each new rectangle (by iterating over ones that it overlaps with, deleting them, and then inserting a new interval spanning the affected x-coordinates), we can also easily update the union's current perimeter along the way. Efficient ordered sets are supported in many programming languages, backed by BBSTs (balanced binary search trees) and capable of supporting the necessary searching, deletion, and insertion operations in logarithmic time in the number of elements in the set. Therefore, given that our ordered set will contain at most \(N\) elements (one interval per rectangle) and that each rectangle's interval will be inserted/deleted at most once, this approach yields a time complexity of \(O(N log(N))\). diff --git a/2020/round1/perimetric_ch3.cpp b/2020/round1/perimetric_ch3.cpp new file mode 100644 index 0000000000000000000000000000000000000000..f33599c7694ed0a25b459cab534ef3bfe62bceca --- /dev/null +++ b/2020/round1/perimetric_ch3.cpp @@ -0,0 +1,157 @@ +// Perimetric - Chapter 3 +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 1000001 +#define MOD 1000000007 + +#define TR pair + +LL cur, ans; +int L[LIM], W[LIM], H[LIM]; +set S; + +void ReadSeq(int* V, int N, int K) +{ + int i, A, B, C, D; + Fox(i, K) + Read(V[i]); + Read(A), Read(B), Read(C), Read(D); + FoxI(i, K, N - 1) + V[i] = ((LL)A * V[i - 2] + (LL)B * V[i - 1] + C) % D + 1; +} + +void ProcessRect(int L, int R, int H) +{ + // tentatively add full rectangle perimeter + cur += 2 * (R - L + H); + // iterate over union height changes overlapping with this rectangle + auto I = prev(S.lower_bound(mp(L, mp(-1, -1)))); + auto J = S.lower_bound(mp(R + 1, mp(-1, -1))); + TR A = *I, B = *J; + while (I->x < B.x) + { + J = I, I++; + if (J->y.y) + cur -= 2 * (min(R, I->x) - max(L, J->x)); // subtract swallowed-up horizontal perimeter + if (J->x >= L) + { + cur -= Abs(J->y.x - J->y.y); // subtract swallowed-up vertical perimeter + S.erase(J); + } + } + cur -= A.y.y + B.y.x; // subtract swallowed-up vertical perimeter at either end of this rectangle + // insert new union height changes at either end of this rectangle + if (A.y.y != H) + S.insert(mp(L, mp(A.y.y, H))); + if (B.y.x != H) + S.insert(mp(R, mp(H, B.y.x))); +} + +int ProcessCase() +{ + int N, K, i; + // input + Read(N), Read(K); + ReadSeq(L, N, K); + ReadSeq(W, N, K); + ReadSeq(H, N, K); + // process rectangles, while maintaining set of union height changes + cur = 0, ans = 1; + S.clear(); + S.insert(mp(-1, mp(0, 0))); + S.insert(mp(INF + 1, mp(0, 0))); + Fox(i, N) + { + ProcessRect(L[i], L[i] + W[i], H[i]); + ans = (LL)ans * (cur % MOD) % MOD; + } + return(ans); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/round1/perimetric_ch3.in b/2020/round1/perimetric_ch3.in new file mode 100644 index 0000000000000000000000000000000000000000..7166ce451fca3e5c40eef3c976362b07eeecbd6f --- /dev/null +++ b/2020/round1/perimetric_ch3.in @@ -0,0 +1,757 @@ +108 +2 2 +1 2 +0 0 0 100 +2 2 +0 0 0 100 +3 3 +0 0 0 100 +2 2 +10 20 +0 0 0 100 +2 2 +0 0 0 100 +3 3 +0 0 0 100 +5 5 +8 3 10 16 2 +0 0 0 100 +3 1 5 2 18 +0 0 0 100 +1 4 6 6 9 +0 0 0 100 +10 3 +14 5 14 +4 7 2 47 +6 4 13 +2 3 17 33 +11 13 14 +1 0 2 31 +50 10 +100 26 4 28 106 59 9 105 10 97 +14 40 784 4832 +130 12 82 487 12 30 214 104 104 527 +21 81 410 605 +38 103 109 109 116 157 181 217 234 234 +1 0 16 951 +350 30 +4308 1398 4618 3986 5882 4096 4539 1916 1081 4194 5961 5977 5894 4292 5933 4741 2834 2888 3759 3856 3392 856 1266 3819 3108 3498 2141 5637 920 5452 +746670471 536724916 890536760 5980 +166 5523 3063 2366 491 3162 2539 3449 5376 813 4708 2441 3178 2241 3497 636 3572 4504 762 594 2979 4871 2891 645 1170 2439 797 5134 696 917 +104430518 804133208 334544093 6570 +19 23 23 32 38 44 50 50 59 64 66 66 75 80 81 87 88 95 102 107 111 119 120 125 128 134 136 139 146 147 +0 1 65 20007246 +500 50 +4577 4109 4695 6970 5557 6921 2252 2153 6219 6262 231 5859 1901 2176 397 7156 6011 4787 162 3275 1771 5028 2264 6776 580 7237 1332 1963 4707 6274 4309 751 5361 6418 6746 3908 1087 2419 2077 4674 7788 5456 4158 1847 2188 3985 7552 2897 3209 964 +658198934 574138063 690901506 7859 +169 468 1212 1199 1115 1031 687 509 862 784 1237 635 1202 869 479 45 523 486 173 1208 258 860 869 616 166 364 1166 1220 1101 864 1047 333 824 361 397 558 11 490 987 667 662 1205 460 168 1056 1112 1208 818 981 1332 +639225282 870924056 491232775 1380 +13 21 25 25 33 36 41 49 50 52 57 65 66 72 80 89 97 100 108 116 117 122 131 132 134 136 143 143 144 144 146 151 157 158 163 170 179 188 189 193 196 199 199 207 216 225 232 239 246 250 +0 1 12 20008202 +200000 2 +1 3 +0 1 100 500000000 +100 100 +0 0 99 500000000 +300000000 300000000 +1 0 0 500000000 +1000000 2 +1 2 +0 1 0 500000000 +2 3 +0 1 0 10000000 +1 2 +0 1 0 500000000 +1000000 2 +1 10000000 +0 1 10000000 500000000 +2 3 +0 1 0 10000000 +1 2 +0 1 0 500000000 +1000000 2 +1 496 +0 1 494 500000000 +494 494 +0 1 9999999 10000000 +499999995 499999995 +0 1 499999999 500000000 +1000000 2 +427153489 472353423 +527695706 320891100 900509661 500000000 +1 1 +1 1 0 10000000 +1 2 +0 1 0 500000000 +22100 2 +499999999 499977899 +0 1 499977900 500000000 +22099 22098 +0 1 499999998 500000000 +1 2 +0 1 0 500000000 +1000000 2 +100000000 100000001 +0 1 0 500000000 +100000000 100000001 +0 1 499999999 500000000 +1 2 +0 1 0 500000000 +1000000 2 +100000000 99999999 +0 1 499999999 500000000 +1 2 +0 1 0 500000000 +1 2 +0 1 0 500000000 +987654 2 +1 2 +21500543 279356914 593559733 500000000 +499999999 85539793 +620990846 839504882 515276028 500000000 +6 14 +0 1 10 500000000 +994912 9949 +485 693 311 132 11 23 64 157 268 649 350 57 498 533 67 130 25 79 15 312 670 240 700 599 332 761 524 543 284 701 427 451 575 677 204 756 648 380 588 429 111 635 411 726 247 128 545 497 645 742 366 249 39 100 711 746 129 421 687 608 652 530 643 478 222 353 374 110 670 626 698 666 703 519 121 224 27 369 378 762 222 675 357 277 361 181 258 115 247 698 159 193 331 217 58 97 63 648 557 297 689 56 460 613 475 651 705 580 748 788 272 659 787 342 296 390 379 454 52 270 351 725 204 530 32 458 396 217 340 136 188 480 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368 +0 1 81 20008551 +481 50 +433 4261 4550 2411 4331 2347 6647 4579 2633 8535 8704 8348 4774 1863 197 4013 8691 7608 6107 3891 6529 3123 6863 8195 7763 2006 6192 3437 4561 5365 560 7288 3470 3358 3518 3218 6280 2698 7883 206 3656 8550 2122 1162 5445 1253 5019 1881 405 4168 +136472996 604139396 742753873 8943 +1308 2613 603 2322 214 877 854 1363 693 2281 219 1445 711 886 1121 1273 1823 357 2235 1771 896 1486 380 399 1638 346 1540 817 2141 2612 653 1581 20 842 1641 1758 498 2469 210 1793 459 1979 602 1087 2177 341 1189 2557 870 713 +823423181 974920705 343840548 2707 +7 8 12 13 16 23 27 35 42 44 49 54 60 68 69 77 83 88 92 94 98 103 107 107 115 120 121 124 133 133 141 150 158 165 170 174 175 177 183 184 184 187 192 192 193 197 199 202 209 210 +0 1 77 20003116 +845 86 +460 181 437 70 435 245 621 328 32 341 198 182 485 157 52 572 274 405 423 279 590 412 55 250 354 90 402 301 254 100 341 358 58 22 1 241 218 512 337 259 264 516 391 51 358 86 501 420 246 220 544 98 570 480 32 449 24 236 539 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591 611 4024 5342 2623 3464 3242 4023 1554 3452 1325 3582 1586 4321 5223 3675 3001 2557 3893 5005 1717 3482 4895 2219 2406 1518 1475 5429 2921 1916 4207 966 +28731286 387457483 34116002 5646 +8 10 12 17 23 27 34 40 43 45 45 48 57 64 68 75 82 86 91 95 103 109 117 123 123 132 134 140 140 148 156 164 172 175 182 183 184 188 197 205 214 222 224 224 232 240 240 247 250 258 266 274 279 284 292 297 306 313 313 322 329 330 330 333 334 337 344 348 354 358 367 367 371 375 384 384 391 +0 1 22 20005274 +673 69 +267 441 2008 1646 1878 1239 1184 3585 702 444 1654 2085 3704 339 3712 2311 3202 1625 1803 1538 1559 1337 1708 3436 828 1752 3839 2988 1704 3685 352 1001 1629 3287 135 3536 2908 91 2077 3198 1935 3141 3444 692 3169 3036 1348 1644 2330 722 1264 1697 881 711 457 1216 1921 2109 3590 2225 1156 980 346 2589 1507 1764 378 1694 3279 +4253155 362502175 965983644 3900 +2150 307 2373 6582 4711 4476 4318 6261 2586 2833 650 6485 610 3776 2788 1237 6410 3665 6009 3382 5542 429 4268 5527 4530 2521 2970 1833 1172 560 1430 812 1914 121 2585 5288 6517 4345 5733 3582 129 3107 4626 3780 5510 762 3849 4005 6720 3864 5999 2193 2023 3613 6068 631 6674 2729 3614 154 3546 5627 4997 966 4551 6365 2318 4334 6805 +554782611 488468875 375597448 6922 +17 18 23 28 37 46 51 60 61 69 69 73 80 81 88 94 99 106 111 118 119 122 123 129 138 139 141 149 157 158 163 171 172 180 180 183 185 194 195 199 201 210 211 214 215 224 233 241 249 254 261 262 264 270 273 273 277 284 291 294 303 308 312 313 317 322 325 333 338 +0 1 40 20000045 +516 53 +526 3932 6495 6573 2514 4150 6410 4995 7520 5916 7365 1997 3374 606 7425 5093 6939 514 7131 5243 5028 3134 5995 3874 1459 1704 1406 2483 7048 3006 6754 5880 1339 6131 2751 318 5012 6150 5887 372 3801 3434 6462 3630 2016 6003 1724 5613 6305 6190 4687 5177 5615 +687936738 195247266 560922563 7653 +4328 5216 3143 2475 3549 2067 4216 5538 5150 2577 4090 2763 2901 963 4344 4399 3872 616 211 3806 3182 5955 3955 4364 1618 301 2277 2302 2880 1744 682 2201 2544 4732 2269 4327 1852 4023 5272 5653 880 4956 4961 527 377 3659 4763 5053 2752 4233 3107 3114 1795 +240732642 685278946 182397676 6349 +3 4 10 18 27 36 43 49 56 59 67 76 77 84 92 97 97 104 109 115 119 126 131 140 148 149 154 160 161 168 175 176 185 192 199 201 202 208 213 218 224 231 232 241 249 249 258 258 261 268 272 280 281 +0 1 28 20007977 +189 20 +4212 3927 3134 409 2196 5984 1906 2721 1938 2577 195 1533 2075 1330 5178 2908 2532 4997 2267 3218 +755090738 857965707 230336425 6229 +44 200 1120 608 396 227 100 346 1331 1080 1167 250 1632 109 852 609 1480 750 1311 107 +973195757 711306891 32820647 1713 +16 24 31 38 41 42 42 44 50 54 55 62 64 69 70 72 74 74 82 91 +0 1 93 20009887 +909 92 +1619 15 1225 293 1668 884 418 618 774 473 250 899 1388 212 274 85 247 1286 1778 1457 726 770 932 486 663 804 1353 1282 1358 1020 1520 307 1180 21 1405 1211 465 506 1739 1675 725 1367 987 1188 1837 1255 185 639 1970 1634 1881 1796 1824 1120 596 1799 754 1177 1441 371 1662 1808 1007 1254 127 778 1977 661 1362 1524 1201 1220 1382 1241 346 614 1694 472 1639 313 519 129 1603 325 1937 958 525 336 1319 352 1475 1258 +287141386 352583479 874228940 1986 +55 83 54 20 87 18 3 22 25 45 43 53 66 5 27 60 11 38 63 41 14 85 61 45 17 85 38 69 10 9 50 8 22 29 57 65 87 76 67 35 49 47 24 72 64 89 79 52 28 18 54 69 3 21 5 41 44 38 89 34 25 22 16 29 2 84 49 17 69 56 65 3 23 25 21 61 69 87 20 28 48 24 86 23 39 50 7 49 14 47 9 19 +610153750 30377898 72820818 92 +1 8 14 14 20 23 30 34 40 45 49 49 53 59 66 67 75 79 83 83 84 87 94 98 106 108 110 119 121 128 136 139 143 146 150 150 159 162 167 173 176 183 187 196 198 207 214 223 229 230 231 239 242 250 253 254 262 267 269 275 278 284 287 296 298 300 300 300 305 311 320 328 333 335 335 341 350 358 364 364 366 375 383 388 393 399 403 404 406 408 415 419 +0 1 58 20005286 diff --git a/2020/round1/perimetric_ch3.md b/2020/round1/perimetric_ch3.md new file mode 100644 index 0000000000000000000000000000000000000000..c92ac19733e552d8df1ca5d126776d1fac26245e --- /dev/null +++ b/2020/round1/perimetric_ch3.md @@ -0,0 +1,65 @@ +**Note: This problem shares similarities with Chapters [1](https://www.facebook.com/codingcompetitions/hacker-cup/2020/round-1/problems/A1) and [2](https://www.facebook.com/codingcompetitions/hacker-cup/2020/round-1/problems/A2). The solution to any chapter may help with solving the others, so please consider reading all of them.** + +After a highly successful haul from the log drive, Connie the contractor is tasked with building a number of houses in the Great White North. For each job, the client has provided a floor plan consisting of \(N\) rectangular rooms, numbered from 1 to \(N\). From a bird's-eye view, the rooms are arranged on a 2-dimensional plane, with axis-aligned walls. The southern wall of each room has y-coordinate 0. + +The \(i\)th rectangular room has southwest corner \((L_i, 0)\) and northeast corner \((L_i + W_i, H_i\)). **In this chapter of the problem, the rooms have non-decreasing \(H\) values \((H_1 \le H_2 \le ... \le H_N)\).** Since houses often have shared regions (such as a common living/dining area), these rooms may overlap with one another. + +Unfortunately, [log houses are quite susceptible to air leakage](https://www.energy.gov/energysaver/types-homes/energy-efficiency-log-homes). Connie knows that she must install additional insulation to keep the houses warm and energy-efficient during the harsh Canadian winters. In order to determine the amount of insulation material required, Connie will first need to gather some metrics: the perimeters around various combinations of rooms. + +Specifically, let \(P_i\) be the perimeter of the union of rooms \(1..i\). Note that any given point is considered to be within the union if and only if it's within at least one of the rooms' rectangles (including right on an edge), and that the union might not form a single connected polygon. Please help compute the product \((P_1 * P_2 * ... * P_N)\). As this product may be very large, you should compute its value modulo 1,000,000,007. + +In order to reduce the size of the input, the rooms' coordinates will not all be provided explicitly. Instead, you'll be given the first \(K\) values \(L_{1..K}\), \(W_{1..K}\), and \(H_{1..K}\), as well as the three quadruples of constants \((A_L, B_L, C_L, D_L)\), \((A_W, B_W, C_W, D_W)\), and \((A_H, B_H, C_H, D_H)\), and must then compute \(L_{(K+1)..N}\), \(W_{(K+1)..N}\), and \(H_{(K+1)..N}\) as follows: + +\(L_i = ((A_L * L_{i-2} + B_L * L_{i-1} + C_L) \text{ modulo } D_L) + 1\) for \(i > K\) +\(W_i = ((A_W * W_{i-2} + B_W * W_{i-1} + C_W) \text{ modulo } D_W) + 1\) for \(i > K\) +\(H_i = ((A_H * H_{i-2} + B_H * H_{i-1} + C_H) \text{ modulo } D_H) + 1\) for \(i > K\) + +# Input + +Input begins with an integer \(T\), the number of floor plans. For each plan, there are 7 lines. + +The first line contains the 2 space-separated integers \(N\) and \(K\). + +The second line contains the \(K\) space-separated integers \(L_{1..K}\). + +The third line contains the 4 space-separated integers \(A_L\), \(B_L\), \(C_L\), and \(D_L\). + +The fourth line contains the \(K\) space-separated integers \(W_{1..K}\). + +The fifth line contains the 4 space-separated integers \(A_W\), \(B_W\), \(C_W\), and \(D_W\). + +The sixth line contains the \(K\) space-separated integers \(H_{1..K}\). + +The seventh line contains the 4 space-separated integers \(A_H\), \(B_H\), \(C_H\), and \(D_H\). + +# Output + +For the \(i\)th floor plan, print a line containing *"Case #i: "* followed by a single integer, the product \((P_1 * P_2 * ... * P_N)\) as defined above, modulo 1,000,000,007. + +# Constraints + +\(1 \le T \le 90\) +\(2 \le N \le 1,000,000\) +\(2 \le K \le N\) +\(0 \le A_L, B_L, C_L, A_W, B_W, C_W, A_H, B_H, C_H \le 1,000,000,000\) +\(1 \le D_L, D_W, D_H \le 500,000,000\) +\(1 \le L_i \le D_L\) +\(1 \le W_i \le D_W\) +\(1 \le H_i \le D_H\) +\(H_1 \le H_2 \le ... \le H_N\) + +The sum of \(N\) across all floor plans is at most 10,000,000. + +# Explanation of Sample + +For the first floor plan, \(L = [1, 2]\) and \(H = [3, 3]\). The union of just the first room's rectangle is equivalent to that rectangle, which has a width of 2 and a height of 3, and therefore a perimeter of 10. The union of both rooms' rectangles is equivalent to a single rectangle with a width of 3 and a height of 3 (with southwest corner (1, 0)), having a perimeter of 12. The answer is therefore \((10 * 12)\text{ modulo }1\text{,}000\text{,}000\text{,}007 = 120\). + +For the second floor plan, \(P = [10, 20]\). + +For the third floor plan, \(P = [8, 18, 36, 52, 54]\). + +For the fourth floor plan: +\(L = [14, 5, 14, 27, 13, 14, 12, 2, 18, 43]\) +\(W = [6, 4, 13, 32, 8, 7, 22, 32, 26, 28]\) +\(H = [11, 13, 14, 16, 17, 19, 20, 22, 23, 25]\) +\(P = [34, 68, 88, 156, 164, 168, 168, 158, 160, 188]\) diff --git a/2020/round1/perimetric_ch3.out b/2020/round1/perimetric_ch3.out new file mode 100644 index 0000000000000000000000000000000000000000..4e9cdede22f6950824a5f24f38f5b57362a0c8cc --- /dev/null +++ b/2020/round1/perimetric_ch3.out @@ -0,0 +1,108 @@ +Case #1: 120 +Case #2: 200 +Case #3: 14556672 +Case #4: 842325442 +Case #5: 848278949 +Case #6: 724024029 +Case #7: 883159606 +Case #8: 341232474 +Case #9: 177384201 +Case #10: 59533825 +Case #11: 471168914 +Case #12: 292642856 +Case #13: 566076553 +Case #14: 985813229 +Case #15: 835077496 +Case #16: 720147266 +Case #17: 343337130 +Case #18: 583164385 +Case #19: 449906586 +Case #20: 173099471 +Case #21: 864641663 +Case #22: 558291542 +Case #23: 438311570 +Case #24: 9881067 +Case #25: 15746517 +Case #26: 839308384 +Case #27: 866590882 +Case #28: 586655890 +Case #29: 319103819 +Case #30: 545824851 +Case #31: 422241659 +Case #32: 568796472 +Case #33: 903428092 +Case #34: 591161754 +Case #35: 68407412 +Case #36: 362774227 +Case #37: 62869318 +Case #38: 821733177 +Case #39: 206663437 +Case #40: 936948902 +Case #41: 870114671 +Case #42: 702878243 +Case #43: 566801503 +Case #44: 994941055 +Case #45: 591695173 +Case #46: 13777874 +Case #47: 475283121 +Case #48: 296867143 +Case #49: 243853867 +Case #50: 284564393 +Case #51: 789954190 +Case #52: 247367076 +Case #53: 226952114 +Case #54: 717460853 +Case #55: 522490645 +Case #56: 174192311 +Case #57: 244967368 +Case #58: 399458736 +Case #59: 117612307 +Case #60: 883990586 +Case #61: 736149684 +Case #62: 9531360 +Case #63: 835811635 +Case #64: 562338795 +Case #65: 325564179 +Case #66: 370130680 +Case #67: 5367876 +Case #68: 97764063 +Case #69: 113617735 +Case #70: 889270738 +Case #71: 377660449 +Case #72: 587368195 +Case #73: 790412854 +Case #74: 534742184 +Case #75: 104133582 +Case #76: 566868542 +Case #77: 638873001 +Case #78: 674772660 +Case #79: 791092996 +Case #80: 772709752 +Case #81: 656649814 +Case #82: 222937741 +Case #83: 228069349 +Case #84: 363313343 +Case #85: 394517232 +Case #86: 806449894 +Case #87: 164853876 +Case #88: 490557845 +Case #89: 176612232 +Case #90: 582757160 +Case #91: 77065332 +Case #92: 987905814 +Case #93: 619762746 +Case #94: 136956176 +Case #95: 135596971 +Case #96: 75745841 +Case #97: 645451645 +Case #98: 472114708 +Case #99: 703853278 +Case #100: 173243597 +Case #101: 424581035 +Case #102: 373683782 +Case #103: 231526635 +Case #104: 177450524 +Case #105: 860053814 +Case #106: 119313600 +Case #107: 591731756 +Case #108: 701841402 diff --git a/2020/round1/perimetric_ch3_sol.md b/2020/round1/perimetric_ch3_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..3e0a09c42d22e9e3de6f903657b62978eeb2d0a3 --- /dev/null +++ b/2020/round1/perimetric_ch3_sol.md @@ -0,0 +1,7 @@ +As in Chapters 1 and 2 of this problem, we'll process the rectangles one by one while computing \(P_{1..N}\). When processing the \(i\)th rectangle, we can't afford to compute the perimeter of the union of the first \(i\) rectangles from scratch (as this would result in an inefficient algorithm with time complexity at least \(O(N^2)\)), so we'll instead compute how the inclusion of this rectangle affects the existing rectangle union (and therefore how \(P_i\) differs from \(P_{i-1}\), assuming that \(P_0 = 0\)). + +Unlike in Chapter 2, the heights of the existing rectangles do matter in this chapter. A convenient way to maintain information about them is with an ordered set of vertical line segments (or "height changes") present in the union's perimeter. Each of these is a tuple \((x, y_1, y_2)\) indicating that, at x-coordinate \(x\), the top portion of the perimeter moves up/down from y-coordinate \(y_1\) to y-coordinate \(y_2\) when tracing the perimeter from left to right. + +When processing the \(i\)th rectangle, we can begin by tentatively adding on its full perimeter (\(2(W_i + H_i)\)) to the union's current perimeter, and then separately consider the amount of horizontal and vertical perimeter we've overcounted. Similarly to Chapter 2, the total horizontal perimeter of the rectangle union is equal to twice the total length of x-coordinates spanned by at least one rectangle, meaning that the amount of overcounted horizontal perimeter is twice the total length of x-coordinates spanned by at least one existing rectangle within the interval \([L_i, L_i + W_i]\). We can tally that up by iterating over the height changes which the \(i\)th rectangle overlaps with, noting that the interval from a height change \((x, y_1, y_2)\) to the next height change is spanned by an existing rectangle if and only if \(y_2 > 0\). As for overcounted vertical perimeter, each existing height change which the \(i\)th rectangle overlaps (inclusively) with is entirely swallowed up and should no longer contribute its vertical length to the union's perimeter. Furthermore, the height of the existing perimeter at x-coordinate \(L_i\) (computable by looking at the nearest height change before or after that x-coordinate) should be subtracted (as the perimeter actually only goes down from the top of the \(i\)th rectangle to that y-coordinate rather than all the way to \(0\)), and the same applies for x-coordinate \(L_i + W_i\). + +Based on the above observations, as long as we're able to update the ordered set of height changes for each new rectangle (by iterating over ones that it overlaps with, deleting them, and then inserting new ones at its left and right ends if appropriate), we can also easily update the union's current perimeter along the way. As in Chapter 2, we'll use an ordered set implementation capable of supporting the necessary searching, deletion, and insertion operations in logarithmic time (using a BBST). Given that our ordered set will contain at most \(2N\) elements (two height changes per rectangle) and that each rectangle's height changes will be inserted/deleted at most once, this approach yields a time complexity of \(O(N log(N))\). diff --git a/2020/round1/quarantine.cpp b/2020/round1/quarantine.cpp new file mode 100644 index 0000000000000000000000000000000000000000..5c0338b490c5cb7dbcd2fcf78f82597e3bf26c03 --- /dev/null +++ b/2020/round1/quarantine.cpp @@ -0,0 +1,219 @@ +// Quarantine +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 1000001 +#define MOD 1000000007 + +#define MCPR pair + +int N, K, A, B; +char S[LIM]; +int E[LIM], C[LIM]; +int pi, P[LIM], L[LIM], R[LIM]; +vector ch[LIM]; +MCPR preM[LIM], sufM[LIM], subM[LIM]; + +void ReadSeq(int* V, int N, int K) +{ + int i, A, B, C; + Fox1(i, K) + Read(V[i]); + Read(A), Read(B), Read(C); + FoxI(i, K + 1, N - 1) + V[i] = ((LL)A * V[i - 2] + (LL)B * V[i - 1] + C) % i + 1; +} + +// traverses the tree in pre-order, numbering nodes and storing their subtrees' ranges +void Pre(int i) +{ + P[pi] = i; + L[i] = pi++; + for (auto j : ch[i]) + Pre(j); + R[i] = pi - 1; +} + +// returns number of "*" nodes reachable from i in its subtree +int CountC(int i) +{ + if (S[i] == '#') + return(0); + int c = 1; + for (auto j : ch[i]) + c += CountC(j); + return(c); +} + +// sets C values of all "*" nodes reachable from i in its subtree +void SetC(int i, int c) +{ + if (S[i] == '#') + return; + C[i] = c; + for (auto j : ch[i]) + SetC(j, c); +} + +// combines two max/count pairs +MCPR Comb(MCPR a, MCPR b) +{ + LL m = max(a.x, b.x); + return(mp(m, (a.x == m ? a.y : 0) + (b.x == m ? b.y : 0))); +} + +MCPR ProcessCase() +{ + int i, j; + // init + Fill(C, 0); + Fox(i, N) + ch[i].clear(); + // input, and generate E values + Read(N), Read(K); + scanf("%s", &S); + ReadSeq(E, N, K); + Fox1(i, N - 1) + { + E[i]--; + ch[E[i]].pb(i); + } + // number nodes in pre-order + pi = 0; + Pre(0); + // floodfill to find connected * regions + int nr = 0; + LL base = 0; + Fox(i, N) + if (S[i] == '*' && !C[i]) + { + nr++; + int c = CountC(i); + SetC(i, c); + base += (LL)c * (c - 1) / 2; + } + // precompute prefix, suffix, and subtree C value max/count pairs + preM[0] = sufM[N] = mp(-1, 0); + Fox(i, N) + preM[i + 1] = Comb(preM[i], mp(C[P[i]], 1)); + FoxR(i, N) + { + sufM[i] = Comb(sufM[i + 1], mp(C[P[i]], 1)); + subM[i] = mp(C[i], 1); + for (auto j : ch[i]) + subM[i] = Comb(subM[i], subM[j]); + } + // consider each possible edge to delete + MCPR ans = mp(-1, 0); + Fox1(i, N - 1) + if (S[i] == '#' || S[E[i]] == '#' || nr == 1) + { + // compute max/count pairs inside/outside i's subtree + MCPR in, out; + if (nr == 1 && (S[i] == '#' || S[E[i]] == '#')) + { + // max values are irrelevant in this case + in = mp(0, R[i] - L[i] + 1); + out = mp(0, N - in.y); + } + else + { + in = subM[i]; + out = Comb(preM[L[i]], sufM[R[i] + 1]); + } + MCPR cur = mp(base, in.y * out.y); + if (nr > 1) + cur.x += in.x * out.x; + ans = Comb(ans, cur); + } + return(ans); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + { + MCPR ans = ProcessCase(); + printf("Case #%d: %lld %lld\n", t, ans.x, ans.y); + } + return(0); +} \ No newline at end of file diff --git a/2020/round1/quarantine.in b/2020/round1/quarantine.in new file mode 100644 index 0000000000000000000000000000000000000000..d7498c9d72a711d8be0fb26a2a26d5832cb150d9 --- /dev/null +++ b/2020/round1/quarantine.in @@ -0,0 +1,473 @@ +118 +3 2 +*** +1 2 +0 0 0 +3 2 +### +1 2 +0 0 0 +3 2 +#** +1 1 +0 0 0 +6 5 +*#***# +1 1 2 2 3 +0 0 0 +5 4 +**##* +1 1 2 2 +0 0 0 +10 2 +**#*#***#* +1 2 +3 4 5 +50 5 +#*##***#**###*#****#***##**#*#**##****#*****#**#** +1 1 3 2 5 +31 4 15 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*********************************##********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************##********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#*****************************************************************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************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************##********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************************************#*********************************************# 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***************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************##***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***********************************************************************************************************************************************************************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***************************************************************************************************************************************************************************##***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************##***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************##***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************##***************************************************************************************************************************************************************************************************************************************#***************************************************************************************************************************************************************************************************************************************#****************************************************************************************************************# 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***********************##********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************##********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************##********************************************************************************************************************************************************************************************##********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#**********************************************************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************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************##********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************##********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************#********************************************************************************************************************************************************************************************##**********************************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*****************************************************************************************#*********************************************************************************************************************************************************##*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************##*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************##*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************##*********************************************************************************************************************************************************#*********************************************************************************************************************************************************##*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************#*********************************************************************************************************************************************************##**********************************************************************************************************************************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+1 2 +0 1 0 +38922 2 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584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 +335149534 517693687 22652983 diff --git a/2020/round1/quarantine.md b/2020/round1/quarantine.md new file mode 100644 index 0000000000000000000000000000000000000000..8dc862b9e4be82b7246cbb8755e97fdb0c2ca0e0 --- /dev/null +++ b/2020/round1/quarantine.md @@ -0,0 +1,68 @@ +Let's give it up for all the health care professionals working on the front lines in the face of the COVID-19 pandemic. Many of them can be found at the Stanford-Sunnyvale Teaching Hospital (SSTH), a facility that's working tirelessly to keep the disease under control. The hospital is already following standard infection prevention and control practices, but Dr. Gregory Klaus, M.D. (one of SSTH's most brilliant doctors) is hoping to further reduce transmissions by making some renovations to the building layout. + +A certain floor of the SSTH facility features \(N\) rooms (numbered from \(1\) to \(N\)) and \(N - 1\) hallways (each running directly between a pair of rooms), such that every room is reachable from every other room through a sequence of hallways. In computer science terms, the floor is organized like a tree rooted at room 1. Specifically, for each room \(i\) aside from the first (\(2 \le i \le N)\), there's a hallway running between it and room \(E_i\) that can be traversed in either direction. + +Each room \(i\) contains either a patient that has been diagnosed with COVID-19 (denoted by \(S_i\) = _"#"_), or otherwise an uninfected patient (denoted by \(S_i\) = _"*"_). + +With the help of the hospital's construction team, Dr. Klaus plans to renovate the floor as follows: close down one existing hallway and then construct a new one between two rooms, such that all rooms on the floor still end up reachable from all other rooms. In doing so, he would like to maximize the number of unordered pairs of uninfected patients who can safely reach each other in the resulting layout. An uninfected patient can safely reach another uninfected patient if their rooms are connected by a sequence of hallways which doesn't pass through any rooms with infected patients. It's possible that Dr. Klaus may elect to close down a hallway and then reconstruct a hallway between that same pair of rooms — the renovation budget needs to go somewhere... + +For example, consider the following floor layout, with rooms 2 and 6 containing infected patients: + +{{PHOTO_ID:729459790952901}} + +Initially, only the uninfected patients in rooms 1 and 3 can safely reach each another. However, if Dr. Klaus were to close down the hallway between rooms 1 and 2 and construct a new one between rooms 1 and 5 (as illustrated below), then patients in any of the three pairs of rooms (1, 3), (1, 5), and (3, 5) would be able to reach one another: + +{{PHOTO_ID:294725001615446}} + +Klaus has just two questions: what is the maximum attainable number of such safely-reachable uninfected patient pairs, and what is the number of distinct ordered pairs of hallway closures/constructions yielding that result? + +In order to reduce the size of the input, the values \(E_{2..N}\) will not all be provided explicitly. Instead, you'll be given the first \(K\) values \(E_{2..(K+1)}\), as well as the 3 constants \(A\), \(B\), and \(C\), and must then compute the remaining values \(E_{(K+2)..N}\) as follows: + +\(E_i = ((A * E_{i-2} + B * E_{i-1} + C) \text{ modulo } (i - 1)) + 1\) for \(i > K+1\) + +# Input + +Input begins with an integer \(T\), the number of hospital floors. For each floor, there are 4 lines. + +The first line contains the 2 space-separated integers \(N\) and \(K\). + +The second line contains the length-\(N\) string \(S_{1..N}\). + +The third line contains the \(K\) space-separated integers \(E_{2..K+1}\). + +The fourth line contains the 3 space-separated integers \(A\), \(B\), and \(C\). + +# Output + +For the \(i\)th hospital floor, print a line containing *"Case #i: "* followed by 2 integers, the maximum attainable number of safely-reachable uninfected patient pairs, and the number of distinct renovation plans yielding that result. + +# Constraints + +\(1 \le T \le 80\) +\(3 \le N \le 1,000,000\) +\(2 \le K \le N-1\) +\(S_i \in \{``*", ``\#"\}\) +\(0 \le A, B, C \le 1,000,000,000\) +\(1 \le E_i \le i-1\) + +The sum of N across all floors is at most 4,000,000. + +# Explanation of Sample + +For the first floor, any of the following four renovation plans would result in every uninfected patient being able to safely reach every other uninfected patient (for a total of 3 pairs): + + - Close hallway 1-2, construct hallway 1-2 + - Close hallway 1-2, construct hallway 1-3 + - Close hallway 2-3, construct hallway 1-3 + - Close hallway 2-3, construct hallway 2-3 + +For the second floor, the same four renovation plans would all result in 0 safely-reachable uninfected patient pairs, while still ensuring that all rooms remain reachable from all other rooms. + +For the third floor, only the following two renovation plans would allow the two uninfected patients to safely reach one another: + + - Close hallway 1-2, construct hallway 2-3 + - Close hallway 1-3, construct hallway 2-3 + +The fourth floor is illustrated above, including one of eight optimal renovation plans for it. + +For the sixth floor, \(E_{2..N} = [1, 2, 2, 4, 3, 6, 4, 8, 5]\). diff --git a/2020/round1/quarantine.out b/2020/round1/quarantine.out new file mode 100644 index 0000000000000000000000000000000000000000..5932645e23a791c94fef0f3d4a4b96bf327caee8 --- /dev/null +++ b/2020/round1/quarantine.out @@ -0,0 +1,118 @@ +Case #1: 3 4 +Case #2: 0 4 +Case #3: 1 2 +Case #4: 3 8 +Case #5: 3 12 +Case #6: 15 16 +Case #7: 193 432 +Case #8: 100 180 +Case #9: 131955 122400 +Case #10: 0 8099820001 +Case #11: 3 8099640007 +Case #12: 1 6 +Case #13: 6 33 +Case #14: 3 40 +Case #15: 3 28 +Case #16: 0 48 +Case #17: 1 43 +Case #18: 1 37 +Case #19: 8091278 538345392 +Case #20: 1 999997000002 +Case #21: 3 3999988 +Case #22: 2809277 475327116 +Case #23: 3381850 56077667800 +Case #24: 3896518 83131495150 +Case #25: 2226321 719013120 +Case #26: 3691592 68614386816 +Case #27: 1517330 30549911273 +Case #28: 3911502 56363730192 +Case #29: 2602909 48965096628 +Case #30: 3396145 87572959808 +Case #31: 1710645 25218656154 +Case #32: 2090640 733033470 +Case #33: 2004310 62127431640 +Case #34: 3632221 107577582530 +Case #35: 1837578 70999375000 +Case #36: 2622329 122699674104 +Case #37: 1778239 50274419881 +Case #38: 1695802 59537274732 +Case #39: 3003 17149 +Case #40: 382375 4278724 +Case #41: 191890 2047490 +Case #42: 0 1605878 +Case #43: 0 169151 +Case #44: 0 41292 +Case #45: 307 728 +Case #46: 110 252 +Case #47: 2806 1620 +Case #48: 2111 1568 +Case #49: 11 60 +Case #50: 1089 5600 +Case #51: 255 528 +Case #52: 19 1620 +Case #53: 34977 538109 +Case #54: 326 686 +Case #55: 48 150 +Case #56: 14227 21125 +Case #57: 89348 45108 +Case #58: 4943 8640 +Case #59: 2727 2916 +Case #60: 135 168 +Case #61: 7 112 +Case #62: 100215 17120 +Case #63: 7388 9120 +Case #64: 33731 27060 +Case #65: 0 145 +Case #66: 90 120 +Case #67: 783 462 +Case #68: 106 5780 +Case #69: 41 1125 +Case #70: 1573 27664 +Case #71: 29 405 +Case #72: 2974 5400 +Case #73: 1 334 +Case #74: 162760 5660 +Case #75: 57 125 +Case #76: 898 660 +Case #77: 11 1128 +Case #78: 3217 2752 +Case #79: 584 2700 +Case #80: 60192 72306 +Case #81: 76 3816 +Case #82: 628 1155 +Case #83: 50 144 +Case #84: 1711 22506 +Case #85: 3301 2090 +Case #86: 7 72 +Case #87: 116 100 +Case #88: 3 78 +Case #89: 7699 6952 +Case #90: 13951 105336 +Case #91: 867 5544 +Case #92: 11141 10200 +Case #93: 56 60 +Case #94: 1847 1368 +Case #95: 759 504 +Case #96: 1690 1560 +Case #97: 314 4158 +Case #98: 2491 2220 +Case #99: 744 1386 +Case #100: 64 144 +Case #101: 6 297 +Case #102: 207 504 +Case #103: 436676 11148 +Case #104: 654 360 +Case #105: 398 14168 +Case #106: 1 798 +Case #107: 108 216 +Case #108: 360006 11788 +Case #109: 39 32 +Case #110: 1030 1496 +Case #111: 13734 8448 +Case #112: 140 200 +Case #113: 215746 34020 +Case #114: 2617 2596 +Case #115: 20712 1600 +Case #116: 189 2376 +Case #117: 1088 1748 +Case #118: 346 35623 diff --git a/2020/round1/quarantine_sol.md b/2020/round1/quarantine_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..53cb8ef46fbef09de80e38d17896e58f99315b30 --- /dev/null +++ b/2020/round1/quarantine_sol.md @@ -0,0 +1,9 @@ +We'll define a "region" as a set of _"*"_ nodes reachable from one another without passing through _"#"_ nodes, and the size of a region as the number of nodes in it. We'll then let \(C_i\) be the size of node \(i\)'s region (with \(C_i = 0\) if \(S_i\) = _"#"_). The regions and \(C\) values may all be computed in \(O(N)\) time using floodfill (DFS). We can also compute the initial number of safely-reachable node pairs — each region having size \(s\) contributes \(s(s-1)/2\) such pairs. + +We'll consider each non-root node \(i\) such that the edge between \(i\) and \(E_i\) will be deleted. For now, we'll ignore the case in which there are fewer than two regions, and return to it later. When there are at least two regions, we should only consider edges that won't break up a region (that is, nodes \(i\) such that \(S_i\) = _"#"_ and/or \(S_{E_i}\) = _"#"_). Let's say that the maximum \(C\) value within \(i\)'s subtree is \(m_1\), and there are \(q_1\) nodes in \(i\)'s subtree with that \(C\) value. Let's similarly define \(m_2\) and \(q_2\) for nodes outside \(i\)'s subtree. An edge insertion can then contribute up to \(m_1 m_2\) additional safely-reachable node pairs on top of the initial number, and there are \(q_1 q_2\) edge insertions for which that's the case. + +Returning to the case in which there's at most one region, essentially the only difference is that we may consider deleting any edge. If we delete one between two _"*"_ nodes (thus breaking up the single region), we must ensure that an edge is then inserted between two nodes with non-zero \(C\) values (very similar to the above, except noting that there will be no additional safely-reachable node pairs on top of the initial number). If we delete a different edge, then *any* edge may be inserted (between a node within \(i\)'s subtree and a node outside it). + +Each of the values required above (such as \(m_1\), \(q_1\), \(m_2\), and \(q_2\)) may be obtained in constant time after a \(O(N)\) precomputation process. In particular, we'll begin by performing a pre-order traversal of the tree while numbering the nodes in that order and storing the range of nodes included in each node's subtree (which must be a contiguous subarray of this ordering). Then, for each prefix and suffix of this ordering, as well as for each subtree's subarray, we'll compute two values: the maximum \(C\) value present amongst those nodes, and the number of those nodes having that \(C\) value. + +As such, the time complexity of this solution comes out to \(O(N)\). diff --git a/2020/round2/1530362313804065.jpg b/2020/round2/1530362313804065.jpg new file mode 100644 index 0000000000000000000000000000000000000000..9ca76b55060697040702e86c894e21f554a3bc04 --- /dev/null +++ b/2020/round2/1530362313804065.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b9774e93e8f12a62af5cea8067581c30f1f46e9ed5c98d707797c0622ef893a3 +size 23599 diff --git a/2020/round2/2720312131621256.jpg b/2020/round2/2720312131621256.jpg new file mode 100644 index 0000000000000000000000000000000000000000..3b386538430c72e536e5062cb3b53475236259d1 --- /dev/null +++ b/2020/round2/2720312131621256.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:53f0658ae030e970352dcd8b898d7a7715ec0fea507c49c641f9b460ba3597cf +size 21721 diff --git a/2020/round2/316300889650982.jpg b/2020/round2/316300889650982.jpg new file mode 100644 index 0000000000000000000000000000000000000000..160ea7c3a009601bcf5fcc396dea6ac4bd923318 --- /dev/null +++ b/2020/round2/316300889650982.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:85aa6fa9ae8e3eb6904b502ccf19ae15c1021674fdda4a38c1bac9882a75c646 +size 20530 diff --git a/2020/round2/3296189243803342.jpg b/2020/round2/3296189243803342.jpg new file mode 100644 index 0000000000000000000000000000000000000000..f47ed1fc769eb4d0f2d4ea4e5d238e51defaffc3 --- /dev/null +++ b/2020/round2/3296189243803342.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:14ee50b2b3ef6b510ac4f8ba0c1ce8a93b9ee3548f2a53a05147c1bfc7fe1e7a +size 24578 diff --git a/2020/round2/896677524192910.jpg b/2020/round2/896677524192910.jpg new file mode 100644 index 0000000000000000000000000000000000000000..874152159f7012c896d55a2fb8e0daf411e27f44 --- /dev/null +++ b/2020/round2/896677524192910.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6ed9ce24067002e986f3d139e622fa88719203666b2aea98ad4a0ae7b21ec90b +size 12378 diff --git a/2020/round2/capastaty.cpp b/2020/round2/capastaty.cpp new file mode 100644 index 0000000000000000000000000000000000000000..b373a73b8a4041c1c753e075cb93bd8cce4edb60 --- /dev/null +++ b/2020/round2/capastaty.cpp @@ -0,0 +1,129 @@ +// Ca-pasta-ty +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 1000001 + +int S[LIM], X[LIM], Y[LIM]; + +void ReadSeq(int* V, int N, int K) +{ + int i, A, B, C, D; + Fox(i, K) + Read(V[i]); + Read(A), Read(B), Read(C), Read(D); + FoxI(i, K, N - 1) + V[i] = ((LL)A * V[i - 2] + (LL)B * V[i - 1] + C) % D; +} + +LL ProcessCase() +{ + int N, K, i; + // input + Read(N), Read(K); + ReadSeq(S, N, K); + ReadSeq(X, N, K); + ReadSeq(Y, N, K); + // aggregate requirements + pair P[2]; + Fox(i, N) + { + int x = X[i] - S[i], y = X[i] + Y[i] - S[i]; + P[0].x += max(0, -y); + P[0].y += max(0, -x); + P[1].x += max(0, x); + P[1].y += max(0, y); + } + // check if satisfiable + Fox(i, 2) + if (P[i].x > P[1 - i].y) + return(-1); + return(max(P[0].x, P[1].x)); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %lld\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/round2/capastaty.in b/2020/round2/capastaty.in new file mode 100644 index 0000000000000000000000000000000000000000..c01834291edb15e912399e17ec610a6f7e7944d7 --- /dev/null +++ b/2020/round2/capastaty.in @@ -0,0 +1,771 @@ +110 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He's become so talented that he's decided to open his own restaurant! The challenge is that restaurants today must follow strict capacity guidelines to reduce the virus's spread. He's got the cooking covered, but will need help optimizing the seating logistics in order to achieve a good trade-off between a booming business and responsible social distancing. + +On a certain day, Pascal's restaurant is divided into \(N\) open seating sections (numbered from \(1\) to \(N\)), with \(S_i\) customers initially sitting in each section \(i\). To keep business strong, but seating capacity manageable, Pascal insists that each section \(i\) must seat between \(X_i\) and \(X_i + Y_i\) customers, inclusive. + +In an attempt to achieve this, he may repeatedly choose a customer to politely (yet forcibly) relocate from their current seating section to a different section of Pascal's choice. + +Customers would be annoyed by too much shuffling about, so Pascal has asked for your help in determining the minimum number of such relocations necessary. If the criteria cannot all be met after any number of relocations, you should report a value of \(-1\) instead. + +In order to reduce the size of the input, the above values will not all be provided explicitly. Instead, you'll be given the first \(K\) values \(S_{1..K}\), \(X_{1..K}\), and \(Y_{1..K}\), as well as the three quadruples of constants \((A_S, B_S, C_S, D_S)\), \((A_X, B_X, C_X, D_X)\), and \((A_Y, B_Y, C_Y, D_Y)\), and must then compute \(S_{(K+1)..N}\), \(X_{(K+1)..N}\), and \(Y_{(K+1)..N}\) as follows: + +\(S_i = (A_S * S_{i-2} + B_S * S_{i-1} + C_S)\text{ modulo }D_S\) for \(i > K\) +\(X_i = (A_X * X_{i-2} + B_X * X_{i-1} + C_X)\text{ modulo }D_X\) for \(i > K\) +\(Y_i = (A_Y * Y_{i-2} + B_Y * Y_{i-1} + C_Y)\text{ modulo }D_Y\) for \(i > K\) + +The sum of \(N\) across all \(T\) days is at most 10,000,000. + +# Input + +Input begins with an integer \(T\), the number of days. For each day, there are 7 lines: + +The first line contains the 2 space-separated integers \(N\) and \(K\). + +The second line contains the \(K\) space-separated integers \(S_{1..K}\). + +The third line contains the 4 space-separated integers \(A_S\), \(B_S\), \(C_S\), and \(D_S\). + +The fourth line contains the \(K\) space-separated integers \(X_{1..K}\). + +The fifth line contains the 4 space-separated integers \(A_X\), \(B_X\), \(C_X\), and \(D_X\). + +The sixth line contains the \(K\) space-separated integers \(Y_{1..K}\). + +The seventh line contains the 4 space-separated integers \(A_Y\), \(B_Y\), \(C_Y\), and \(D_Y\). + +# Output + +For the \(i\)th day, print a line containing *"Case #i: "* followed by a single integer, either the minimum number of customer relocations required, or \(-1\) if it's impossible. + +# Constraints + +\(1 \le T \le 100\) +\(2 \le N \le 1,000,000\) +\(2 \le K \le N\) +\(0 \le A_S, B_S, C_S, A_X, B_X, C_X, A_Y, B_Y, C_Y \le 1,000,000,000\) +\(1 \le D_S, D_X, D_Y \le 1,000,000,000\) +\(0 \le S_i < D_S\) +\(0 \le X_i < D_X\) +\(0 \le Y_i < D_Y\) + +# Explanation of Sample + +On the first day, Pascal divides his restaurant into 2 seating sections, with \(S = [10, 20]\), \(X = [9, 21]\), and \(Y = [0, 0]\). In other words, the first section initially has 10 customers but needs to have exactly 9, while the second initially has 20 but needs to have exactly 21. Relocating a single customer from the first section to the second would satisfy both requirements. + +On the second day, each section needs to end up with anywhere between 14 and 18 customers, inclusive. Relocating 4 customers from the second section to the first would accomplish this. + +On the third day, it's impossible to satisfy all of the requirements. For example, if 2 customers were relocated from the first section to the second, the first section would be left with fewer than its required minimum of 2 customers. + +On the fifth day, \(S = [15, 12, 2, 19, 3]\), \(X = [1, 9, 3, 1, 3]\), and \(Y = [0, 6, 9, 9, 8]\). diff --git a/2020/round2/capastaty.out b/2020/round2/capastaty.out new file mode 100644 index 0000000000000000000000000000000000000000..92e27871061c691957e87a2c4c745d8e631fbbc0 --- /dev/null +++ b/2020/round2/capastaty.out @@ -0,0 +1,110 @@ +Case #1: 1 +Case #2: 4 +Case #3: -1 +Case #4: 7 +Case #5: -1 +Case #6: 5172260357 +Case #7: 0 +Case #8: 175394700 +Case #9: -1 +Case #10: 999999 +Case #11: 0 +Case #12: 0 +Case #13: 499999999000000 +Case #14: 124999750000 +Case #15: -1 +Case #16: -1 +Case #17: -1 +Case #18: 12308870560 +Case #19: -1 +Case #20: 11882489538 +Case #21: 159471564 +Case #22: -1 +Case #23: -1 +Case #24: 9636269490 +Case #25: 15352944649 +Case #26: 32649787784 +Case #27: -1 +Case #28: -1 +Case #29: -1 +Case #30: 16118694208 +Case #31: -1 +Case #32: 254397027940 +Case #33: -1 +Case #34: 19241542534 +Case #35: -1 +Case #36: 387532800 +Case #37: -1 +Case #38: -1 +Case #39: 5568440256 +Case #40: -1 +Case #41: -1 +Case #42: -1 +Case #43: 6767739420 +Case #44: -1 +Case #45: 1019482112 +Case #46: -1 +Case #47: 124122421497 +Case #48: 4939100868 +Case #49: -1 +Case #50: -1 +Case #51: 1205904 +Case #52: 1456004160 +Case #53: -1 +Case #54: 14382568255 +Case #55: -1 +Case #56: -1 +Case #57: -1 +Case #58: 9131151120 +Case #59: -1 +Case #60: -1 +Case #61: 4937511852 +Case #62: -1 +Case #63: -1 +Case #64: -1 +Case #65: -1 +Case #66: 14747973586 +Case #67: 0 +Case #68: 147368003994 +Case #69: -1 +Case #70: 42362663616 +Case #71: -1 +Case #72: -1 +Case #73: 34864247904 +Case #74: -1 +Case #75: -1 +Case #76: 50201788 +Case #77: -1 +Case #78: 5453741304 +Case #79: -1 +Case #80: 192885975043 +Case #81: 243904596 +Case #82: -1 +Case #83: -1 +Case #84: 4752034225 +Case #85: 8895905382 +Case #86: -1 +Case #87: -1 +Case #88: 1196827800 +Case #89: -1 +Case #90: -1 +Case #91: 1660660430 +Case #92: -1 +Case #93: -1 +Case #94: 463713152 +Case #95: -1 +Case #96: 6386688 +Case #97: -1 +Case #98: -1 +Case #99: 5431262265 +Case #100: 31824873432 +Case #101: -1 +Case #102: -1 +Case #103: -1 +Case #104: -1 +Case #105: 20181388656 +Case #106: -1 +Case #107: -1 +Case #108: -1 +Case #109: -1 +Case #110: -1 diff --git a/2020/round2/capastaty_sol.md b/2020/round2/capastaty_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..73ac7759f2d7d78d1661b461657577ad2657e3a7 --- /dev/null +++ b/2020/round2/capastaty_sol.md @@ -0,0 +1,13 @@ +For each seating section \(i\), we can compute 4 quantities: + +1. **At least** \(\text{max}(0, X_i - S_i)\) customers must be added to it +2. **At most** \(\text{max}(0, (X_i + Y_i) - S_i)\) customers may be added to it +3. **At least** \(\text{max}(0, S_i - (X_i + Y_i))\) customers must be removed from it +4. **At most** \(\text{max}(0, S_i - X_i)\) customers may be removed from it + +Note that no section will require customers to both be added and removed from it (quantities 1 and 3 cannot both be positive). + +Let \(Q_i\) be the sum of the \(i\)th quantity described above across all \(N\) seating sections. We can compute \(Q_{1..4}\) in \(O(N)\) time. If \(Q_1\) (the minimum total number of customers who must be added to sections) is greater than \(Q_4\) (the maximum total number of customers who may be removed from sections), then the output must be \(-1\). The same applies if \(Q_3 > Q_2\). + +Otherwise, we can observe that it's always possible to relocate customers such that all of the sections' requirements are satisfied, and in particular to do so using only \(\text{max}(Q_1, Q_3)\) relocations (it cannot possibly take fewer). In particular, we can begin by relocating \(\text{min}(Q_1, Q_3)\) customers from sections which must have customers removed into ones which must have customers added. After that, there may either be remaining sections requiring more customers or ones requiring fewer customers, which can be addressed with another \(\text{max}(Q_1, Q_3) - \text{min}(Q_1, Q_3)\) relocations. + diff --git a/2020/round2/circular_circles.cpp b/2020/round2/circular_circles.cpp new file mode 100644 index 0000000000000000000000000000000000000000..0a33f9d17abad86c526f78b3ffea3e3bd628b9ba --- /dev/null +++ b/2020/round2/circular_circles.cpp @@ -0,0 +1,183 @@ +// Circular Circles +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) { + char c, r = 0, n = 0; + x = 0; + for (;;) { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 1000001 +#define MOD 1000000007 + +int N, M, E, K; +int X[LIM], Y[LIM], I[LIM], W[LIM], EW[LIM]; + +LL sumR; +multiset CS[LIM]; // CS[i] = edge weights in circle i +multiset HS[LIM][2]; // HS[i][j] = edge weights in half j of circle i +multiset DS; // DS = circles' D values +multiset IS; // IS = inter-circle edge weights + +void ReadSeq(int* V, int N, int K, int mod) +{ + int i, A, B, C; + Fox(i, K) + Read(V[i]); + Read(A), Read(B), Read(C); + FoxI(i, K, N - 1) + V[i] = ((LL)A * V[i - 2] + (LL)B * V[i - 1] + C) % mod; +} + +void UpdateSet(multiset& S, int v1, int v2) +{ + assert(*S.find(v1) == v1); + S.erase(S.find(v1)); + S.insert(v2); +} + +int GetCircleR(int i) { return(*CS[i].rbegin()); } + +int GetCircleD(int i) { return(*HS[i][0].rbegin() + *HS[i][1].rbegin() - *CS[i].rbegin()); } + +int GetHalf(int i, int j) { return(X[i] <= j && j < Y[i] ? 0 : 1); } + +void UpdateCircleEdge(int i, int j, int w1, int w2) +{ + int r1 = GetCircleR(i), d1 = GetCircleD(i); + UpdateSet(CS[i], w1, w2); + UpdateSet(HS[i][GetHalf(i, j)], w1, w2); + int r2 = GetCircleR(i), d2 = GetCircleD(i); + sumR += r2 - r1; + UpdateSet(DS, d1, d2); +} + +int ProcessCase() +{ + int i, j; + LL sumW; + // init + DS.clear(), IS.clear(); + Fox(i, N) + CS[i].clear(), HS[i][0].clear(), HS[i][1].clear(); + // input + Read(N), Read(M), Read(E), Read(K); + ReadSeq(X, N, K, M); + ReadSeq(Y, N, K, M); + ReadSeq(I, E, K, N * M + N); + ReadSeq(W, E, K, INF); + // initialize edge info + sumW = N * M + N; + sumR = N; + Fox(i, N * M + N) + EW[i] = 1; + Fox(i, N) + { + if (X[i] > Y[i]) + swap(X[i], Y[i]); + Fox(j, 2) + HS[i][j].insert(0); + Fox(j, M) + { + CS[i].insert(1); + HS[i][GetHalf(i, j)].insert(1); + } + DS.insert(GetCircleD(i)); + IS.insert(1); + } + // process events + int ans = 1; + Fox(i, E) + { + j = I[i]; + int w1 = EW[j], w2 = W[i]; + EW[j] = w2; + sumW += w2 - w1; + if (j < N * M) + UpdateCircleEdge(j / M, j % M, w1, w2); + else + UpdateSet(IS, w1, w2); + LL cur = sumW - sumR - max(*DS.rbegin(), *IS.rbegin()); + ans = ans * (cur % MOD) % MOD; + } + return(ans); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/round2/circular_circles.in b/2020/round2/circular_circles.in new file mode 100644 index 0000000000000000000000000000000000000000..e8247a6b51e6e4a571e4eb0c350205905698574c --- /dev/null +++ b/2020/round2/circular_circles.in @@ -0,0 +1,982 @@ +109 +3 3 3 3 +0 2 1 +0 0 0 +1 2 0 +0 0 0 +9 10 11 +0 0 0 +1 1 1 +0 0 0 +3 3 8 3 +0 2 1 +0 0 0 +1 2 0 +0 0 0 +6 10 2 +1 2 3 +0 2 2 +1 0 1 +3 4 10 3 +1 2 0 +0 0 0 +1 1 0 +0 0 0 +14 12 1 +0 1 2 +11 0 6 +0 1 3 +5 6 40 2 +4 1 +3 2 1 +0 3 +1 2 3 +27 14 +4 5 6 +3192874 490177205 +6 5 4 +20 8 100 2 +6 2 +31 92 44 +6 7 +61 40 16 +43 12 +28 6 52 +47 66 +91 86 28 +5 50 500 2 +21 15 +676967874 271827993 346978651 +4 42 +839278050 675804074 205269151 +38 242 +0 1 2 +0 0 +1000000000 1000000000 1000000000 +10 10 100 2 +4 9 +0 1 2 +8 3 +10 11 12 +1 2 +0 1 1 +100 100 +1 0 0 +100 50 500 3 +31 1 27 +49810008 74929960 71181112 +29 12 2 +348577716 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100644 index 0000000000000000000000000000000000000000..ddacc64e444f814e72cd3fcd109206c9ea6da985 --- /dev/null +++ b/2020/round2/circular_circles.md @@ -0,0 +1,105 @@ +Cyrus has come across a curious sight — an undirected, weighted graph with \(N*M\) nodes (numbered from \(0\) to \( N*M - 1\)) and \(N*M + N\) edges (numbered from \(0\) to \(N*M + N - 1\)). Each edge initially has a weight of \(1\). + +What makes this graph so curious is its shape. The nodes are arranged in \(N\) "circles" of \(M\) nodes each. Nodes \(0..(M - 1)\) are in the first circle, nodes \(M..(2*M - 1)\) are in the second, and so on. In each circle, the nodes are arranged in clockwise order (ordered by increasing node number). + +The first \(N*M\) edges lie within circles, with \(M\) edges per circle. Edges \(0..(M - 1)\) lie within the first circle, edges \(M..(2*M - 1)\) lie within the second, and so on. In each circle, its first (lowest-numbered) edge connects the circle's first and second nodes, its second edge connects its second and third nodes, and so on, with its final edge connecting its final and first nodes. + +There are \(N\) more edges (edges \((N*M)..(N*M + N - 1)\)) connecting the \(N\) circles into what might be considered an additional circle. Each circle \(i\) has two "connection points", the first of which is its \((X_i+1)\)st node (that is, node \(M*i + X_i\)), and the second of which is its \((Y_i+1)\)st node (node \(M*i + Y_i\)). \(X_i\) and \(Y_i\) are not necessarily distinct. The \(i\)th of the \(N\) final edges connects the second connection point of circle \(i\) and the first connection point of circle \(((i+1)\text{ modulo }N)\). That is, there's an edge between nodes \(M*0 + Y_0\) and \(M*1 + X_1\), another between \(M*1 + Y_1\) and \(M*2 + X_2\), and so on, with the final edge connecting nodes \(M*(N-1) + Y_{N-1}\) and \(M*0 + X_0\). + +For example, if \(N = 3\), \(M = 3\), \(X = [0, 2, 1]\), and \(Y = [1, 2, 0]\), the graph would look as follows (with the first \(N*M\) edges indicated in blue, and the last \(N\) in red): + +{{PHOTO_ID:316300889650982}} + +Cyrus was hoping to practice computing the [minimum spanning tree](https://en.wikipedia.org/wiki/Minimum_spanning_tree) of this graph, but it's starting to change before his very eyes! \(E\) events are about to occur on the graph (numbered from \(0\) to \(E-1\)), one after another. Event \(i\) will cause the weight of edge \(I_i\) to become \(W_i\). When an edge's weight changes, it will remain at its new value for future events (until changed again). + +Let \(V_i\) be the weight of the graph's minimum spanning tree after the first \(i\) events. Help Cyrus compute the product of \(V_{1..E}\). As this product may be very large, you should only compute its value modulo 1,000,000,007. + +In order to reduce the size of the input, the above values will not all be provided explicitly. Instead, you'll be given the first \(K\) values \(X_{0..(K-1)}\), \(Y_{0..(K-1)}\), \(I_{0..(K-1)}\), and \(W_{0..(K-1)}\), as well as the four triples of constants \((A_X, B_X, C_X)\), \((A_Y, B_Y, C_Y)\), \((A_I, B_I, C_I)\), and \((A_W, B_W, C_W)\), and must then compute \(X_{K..(N-1)}\), \(Y_{K..(N-1)}\), \(I_{K..(E-1)}\), and \(W_{K..(E-1)}\) as follows: + +\(X_i = (A_X * X_{i-2} + B_X * X_{i-1} + C_X)\text{ modulo }M\) for \(i \ge K\) +\(Y_i = (A_Y * Y_{i-2} + B_Y * Y_{i-1} + C_Y)\text{ modulo }M\) for \(i \ge K\) +\(I_i = (A_I * I_{i-2} + B_I * I_{i-1} + C_I)\text{ modulo }(N*M + N)\) for \(i \ge K\) +\(W_i = (A_W * W_{i-2} + B_W * W_{i-1} + C_W)\text{ modulo }1,000,000,000\) for \(i \ge K\) + + +# Input + +Input begins with an integer \(T\), the number of graphs. For each graph, there are 9 lines: + +The first line contains the 4 space-separated integers \(N\), \(M\), \(E\), and \(K\). + +The second line contains the \(K\) space-separated integers \(X_{0..K-1}\). + +The third line contains the 3 space-separated integers \(A_X\), \(B_X\), and \(C_X\). + +The fourth line contains the \(K\) space-separated integers \(Y_{0..K-1}\). + +The fifth line contains the 3 space-separated integers \(A_Y\), \(B_Y\), and \(C_Y\). + +The sixth line contains the \(K\) space-separated integers \(I_{0..K-1}\). + +The seventh line contains the 3 space-separated integers \(A_I\), \(B_I\), and \(C_I\). + +The eighth line contains the \(K\) space-separated integers \(W_{0..K-1}\). + +The ninth line contains the 3 space-separated integers \(A_W\), \(B_W\), and \(C_W\). + +# Output + +For the \(i\)th graph, print a line containing *"Case #i: "* followed by a single integer, the product of \(V_{1..E}\) modulo 1,000,000,007. + +# Constraints + +\(1 \le T \le 90\) +\(3 \le N, M \le 1,000,000\) +\(N*M + N \le 1,000,000\) +\(2 \le E \le 1,000,000\) +\(2 \le K \le min(N, E)\) +\(0 \le A_X, B_X, C_X, A_Y, B_Y, C_Y, A_I, B_I, C_I, A_W, B_W, C_W \le 1,000,000,000\) +\(0 \le X_i, Y_i < M\) +\(0 \le I_i < N*M + N\) +\(0 \le W_i < 1,000,000,000\) + +The sum of \(N*M + E\) across all \(T\) graphs is at most 20,000,000. + +# Explanation of Sample + +**For the first graph:** + +- \(X = [0, 2, 1]\) +- \(Y = [1, 2, 0]\) +- \(I = [9, 10, 11]\) +- \(W = [1, 1, 1]\) + +Each event causes an edge weight to remain equal to \(1\). Therefore, the graph looks as follows, both initially and after all three events: + +{{PHOTO_ID:2720312131621256}} + +A minimum spanning tree on this graph must involve 8 weight-1 edges. Therefore, \(V = [8, 8, 8]\), and the final answer is \((8 * 8 * 8)\text{ modulo 1,000,000,007} = 512\). + +**For the second graph:** + +- \(X = [0, 2, 1]\) +- \(Y = [1, 2, 0]\) +- \(I = [6, 10, 2, 5, 3, 2, 10, 1]\) +- \(W = [0, 2, 2, 3, 3, 4, 4, 5]\) + +After the eighth and final event, the graph looks as follows, with one possible minimum spanning tree highlighted: + +{{PHOTO_ID:1530362313804065}} + +For this graph, \(V = [7, 7, 7, 7, 9, 9, 9, 12]\). + +**For the third graph:** + +- \(X = [1, 2, 0]\) +- \(Y = [1, 1, 0]\) +- \(I = [14, 12, 1, 3, 5, 7, 9, 11, 13, 0]\) +- \(W = [11, 0, 6, 9, 12, 15, 18, 21, 24, 27]\) + +For this graph, \(V = [11, 10, 10, 15, 15, 25, 25, 42, 53, 61]\). + +**For the fourth graph:** + +- \(X = [4, 1, 3, 4, 0]\) +- \(Y = [0, 3, 3, 0, 0]\) diff --git a/2020/round2/circular_circles.out b/2020/round2/circular_circles.out new file mode 100644 index 0000000000000000000000000000000000000000..ba92f0121609d59e28e3ff1848295d104766ed47 --- /dev/null +++ b/2020/round2/circular_circles.out @@ -0,0 +1,109 @@ +Case #1: 512 +Case #2: 21003948 +Case #3: 396727972 +Case #4: 166299874 +Case #5: 849275180 +Case #6: 0 +Case #7: 294032380 +Case #8: 796456345 +Case #9: 715191229 +Case #10: 242746563 +Case #11: 774220861 +Case #12: 510029534 +Case #13: 637180403 +Case #14: 901645920 +Case #15: 539110298 +Case #16: 731957228 +Case #17: 273613783 +Case #18: 123024587 +Case #19: 44067557 +Case #20: 122914388 +Case #21: 87053117 +Case #22: 905758912 +Case #23: 175941257 +Case #24: 705194838 +Case #25: 756899956 +Case #26: 668011195 +Case #27: 85619190 +Case #28: 427852657 +Case #29: 126787535 +Case #30: 792996579 +Case #31: 575732908 +Case #32: 909708555 +Case #33: 356980127 +Case #34: 330326318 +Case #35: 317057027 +Case #36: 531236470 +Case #37: 963519448 +Case #38: 476112727 +Case #39: 622697334 +Case #40: 419873534 +Case #41: 929433169 +Case #42: 899632009 +Case #43: 850676519 +Case #44: 823702088 +Case #45: 314351226 +Case #46: 157473736 +Case #47: 420262624 +Case #48: 781889509 +Case #49: 995084196 +Case #50: 188121137 +Case #51: 896739134 +Case #52: 815999421 +Case #53: 553061310 +Case #54: 626982321 +Case #55: 821955127 +Case #56: 130504928 +Case #57: 972798062 +Case #58: 679132390 +Case #59: 489630481 +Case #60: 550782882 +Case #61: 428899533 +Case #62: 312794194 +Case #63: 682522958 +Case #64: 323577399 +Case #65: 628274590 +Case #66: 665450369 +Case #67: 882484874 +Case #68: 690657728 +Case #69: 278778174 +Case #70: 223997110 +Case #71: 837162234 +Case #72: 782062742 +Case #73: 403504250 +Case #74: 132950209 +Case #75: 899703070 +Case #76: 104495356 +Case #77: 398364871 +Case #78: 408628281 +Case #79: 22565581 +Case #80: 675401981 +Case #81: 639389470 +Case #82: 859145988 +Case #83: 922276869 +Case #84: 932135994 +Case #85: 547993542 +Case #86: 567952361 +Case #87: 130550172 +Case #88: 75730821 +Case #89: 198635873 +Case #90: 953658866 +Case #91: 851760825 +Case #92: 262622474 +Case #93: 926929194 +Case #94: 893611506 +Case #95: 855817242 +Case #96: 460053524 +Case #97: 869672197 +Case #98: 707213320 +Case #99: 90217302 +Case #100: 676599448 +Case #101: 724235412 +Case #102: 792299448 +Case #103: 773710199 +Case #104: 137898020 +Case #105: 538910223 +Case #106: 324930780 +Case #107: 306595544 +Case #108: 720499692 +Case #109: 753030073 diff --git a/2020/round2/circular_circles_sol.md b/2020/round2/circular_circles_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..5ab8979cab67e96cf639a941498a803b0a315aae --- /dev/null +++ b/2020/round2/circular_circles_sol.md @@ -0,0 +1,26 @@ +We can observe that the weight of the minimum spanning tree is equal to the sum of all edge weights in the graph (*Quantity 1*), minus the weight of one removed edge per circle (*Quantity 2*), minus the weight of one additional removed edge somewhere in the graph (*Quantity 3*). + +For *Quantity 3*, the additional removed edge may either be any inter-circle edge, or *potentially* a second edge within some circle. There are two categories of circles: + +- If \(X_i = Y_i\), no two of its edges can be validly removed (without disconnecting the graph). +- Otherwise, if \(X_i \ne Y_i\), the circle can be considered to have two "halves" (the paths connecting its \(X_i\)th and \(Y_i\)th nodes clockwise and counterclockwise), such that up to one edge may be validly removed from *each* half (without disconnecting the graph). + +For each circle \(i\), we'll define three values of interest: + +- Let \(R_i\) be the maximum weight of any edge within the circle. This will help with computing *Quantity 2*. +- Let \(P_i\) be the maximum possible combined weight of any validly-removable pair of edges within the circle (if any). If \(X_i = Y_i\) (meaning that only one of its edges may be removed), we'll let \(P_i = R_i\) instead. +- Let \(D_i = P_i - R_i\). The represents the additional amount by which the minimum spanning tree's weight can be decreased if two edges are removed from the circle, and will help with computing *Quantity 3*. + +Then, for each circle \(i\), we'll maintain: + +- An ordered multiset of all edge weights within it (used to compute \(R_i\)). +- If \(X_i \ne Y_i\), two additional ordered multisets containing all edge weights in each of its halves (used to compute \(D_i\)). + +On top of that, we'll globally maintain: + +- The sum of all edge weights in the graph (*Quantity 1*). +- The sum of all circles' \(R_i\) values (equivalent to *Quantity 2*). +- An ordered multiset of all circles' \(D_i\) values, and another containing all inter-circle edge weights (combined to compute *Quantity 3*). + +Each event (and initial edge) requires us to update just a constant number of the above values and multisets, each taking logarithmic time (provided that the multisets are implemented using BBSTs). Therefore, this solution has a time complexity of \(O((NM + E) (log(N) + log(M)))\). + diff --git a/2020/round2/elimination.cpp b/2020/round2/elimination.cpp new file mode 100644 index 0000000000000000000000000000000000000000..4576576f68281006db865145d308cbaac0bc4cda --- /dev/null +++ b/2020/round2/elimination.cpp @@ -0,0 +1,120 @@ +// Elimination +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +#define LIM 8001 + +// dyn[b][w] = expected number of remaining rounds +// when there are b better and w worse players than you +LD dyn[LIM][LIM]; + +int NumPairs(int c) +{ + return(c * (c - 1) / 2); +} + +// inv indicates that b/w have been swapped to capture worse players winning +LD Comp(int b, int w, bool inv) +{ + LD p = (LD)1 / NumPairs(b + w + 1), d = 0; + // elimination of better player than you + if (b > 0) + d += NumPairs(b) * p * (inv ? dyn[w][b - 1] : dyn[b - 1][w]); + // elimination of worse player than you + if (w > 0) + d += (w * (b + 1) + NumPairs(w)) * p * (inv ? dyn[w - 1][b] : dyn[b][w - 1]); + return(d); +} + +void ProcessCase() +{ + int N, i, b, w; + LD P; + // input + scanf("%d%Lf", &N, &P); + // DP + Fill(dyn, 0); + Fox(b, N) + { + Fox(w, N - b) + { + if (b + w) + { + dyn[b][w] = 1 + + Comp(b, w, false) * P + // better players winning + Comp(w, b, true) * (1 - P); // worse players winning + } + } + } + // output + Fox(i, N) + printf("%.8Lf\n", dyn[N - i - 1][i]); +} + +int main() +{ + int T, t; + scanf("%d\n", &T); + Fox1(t, T) + { + printf("Case #%d:\n", t); + ProcessCase(); + } + return(0); +} \ No newline at end of file diff --git a/2020/round2/elimination.in b/2020/round2/elimination.in new file mode 100644 index 0000000000000000000000000000000000000000..b01358071a30988eece332c9927515dd03a4ea18 --- /dev/null +++ b/2020/round2/elimination.in @@ -0,0 +1,111 @@ +110 +2 1.0 +3 1.0 +4 1.0 +3 0.5 +5 0.678 +9 0.87 +2 0.6 +20 0.75 +200 0.999 +2000 0.5 +8000 1.000 +8000 0.500 +8000 0.684 +2 1.000 +2 0.500 +2 0.684 +2788 0.719 +3984 0.930 +4664 0.797 +4242 0.705 +3127 0.873 +5 0.615 +81 0.743 +27 0.504 +64 0.828 +94 0.726 +16 0.599 +64 0.833 +26 0.638 +4 0.949 +7 0.505 +38 0.515 +41 0.968 +49 0.904 +2 0.574 +100 0.586 +11 0.501 +94 0.561 +16 0.975 +82 0.985 +20 0.591 +13 0.720 +44 0.543 +43 0.824 +63 0.577 +11 0.872 +100 0.685 +81 0.825 +10 0.572 +62 0.646 +68 0.579 +35 0.934 +40 0.553 +37 0.795 +82 0.993 +91 0.907 +66 0.532 +84 0.960 +62 0.790 +9 0.593 +42 0.533 +10 0.600 +85 0.725 +41 0.797 +84 0.983 +56 0.818 +26 0.509 +37 0.853 +43 0.710 +101 0.926 +23 0.974 +68 0.586 +24 0.975 +65 0.582 +44 0.828 +31 0.837 +37 0.684 +59 0.759 +98 0.790 +57 0.884 +92 0.806 +91 0.736 +88 0.863 +67 0.886 +24 0.887 +9 0.938 +91 0.844 +29 0.711 +98 0.752 +65 0.974 +82 0.915 +17 0.882 +38 0.964 +47 0.580 +82 0.520 +53 0.728 +36 0.762 +75 0.551 +67 0.754 +28 0.786 +77 0.676 +65 0.686 +33 0.720 +98 0.779 +87 0.670 +46 0.816 +27 0.517 +21 0.637 +17 0.511 +9 0.592 diff --git a/2020/round2/elimination.md b/2020/round2/elimination.md new file mode 100644 index 0000000000000000000000000000000000000000..377dadbdd3815b4300ceb602b0076b15e6cd34ee --- /dev/null +++ b/2020/round2/elimination.md @@ -0,0 +1,37 @@ +The Facebook Whacker Cup is a series of ping-pong tournaments to find the best paddlers on the social network. A tournament consists of \(N\) ping-pong players and \(N - 1\) elimination matches among them. The players are numbered from \(1\) to \(N\) in increasing order of skill. + +In each match, two players will be randomly selected from the set of players still in contention (if there are \(x\) players, then all \(x(x-1)/2\) unordered pairs of them are equally likely to be selected). Those players will then play against each other, with the more skilled (higher-numbered) of the two winning with a probability of \(P\). The loser will then be eliminated from the tournament, and will not be involved in any future matches. After \(N - 1\) such matches, there will be exactly one player remaining who will be declared the Whacker Cup champion! + +For each player, determine the expected number of matches which will occur while they're still in contention (before they're either eliminated or the tournament concludes). This should include matches which they either directly compete in (including the one in which they lose and get eliminated), or which simply occur before they've been eliminated. Each of your answers must have at most \(10^{-6}\) absolute or relative error to be considered correct. + +# Input + +Input begins with an integer \(T\), the number of tournaments. For each tournament, there is one line containing the space-separated integer \(N\) and real number \(P\) (given with at most 3 digits after the decimal point). + +# Output + +For the \(i\)th tournament, output a line containing *"Case #i:"* followed by \(N\) lines, the \(j\)th of which contains a single real number, the expected number of matches which will occur while player \(j\) is still in contention. + +# Constraints + +\(1 \le T \le 95\) +\(2 \le N \le 8000\) +\(0.5 \le P \le 1.0\) + +The sum of \(N\) across all \(T\) tournaments is at most 50,000. + +# Explanation of Sample + +In the first tournament, player 2 is guaranteed to defeat player 1. Regardless of the result, however, both players will be present for the tournament's single match. + +In the second tournament, player 3 is guaranteed to defeat players 1 and 2, while player 2 is guaranteed to defeat player 1. There are three possible, equally-likely sequences of matches (with `A>B` representing a match in which player \(A\) beats player \(B\)): + +- `2>1` followed by `3>2` +- `3>1` followed by `3>2` +- `3>2` followed by `3>1` + +All players are guaranteed to be in contention during at least the first match. Player 1 has a \(\frac{1}{3}\) chance to also be present for the second match (if the last sequence occurs), player 2's chance is \(\frac{2}{3}\) (if either of the first two sequences occurs), and player 3 is guaranteed to be present for both matches. + +In the third tournament, the match results are again deterministic. The first match will feature one of 6 possible player pairs, the second match will feature one of 3 possible pairs of remaining players, and the third match will be between the final pair of players. Overall, there are \(6 * 3 * 1 = 18\) possible scenarios. Across these scenarios, the total counts of hypothetical matches before elimination are 30 for player 1, 35 for player 2, 43 for player 3, and 54 for player 4. Dividing each by 18 yields the expected answer. + +In the fourth tournament, matches could go either way with equal probability (skill levels don't matter). There are 6 possible, equally-likely first match outcomes (`1>2`, `2>1`, `1>3`, `3>1`, `2>3`, `3>2`), after which only the eliminated player won't be present for the second match. Therefore, each player has a \(\frac{2}{6}\) chance of being present for one match, and a \(\frac{4}{6}\) chance of being present for both. diff --git a/2020/round2/elimination.out b/2020/round2/elimination.out new file mode 100644 index 0000000000000000000000000000000000000000..ad035db4f605a6500e63f32fc638739b173583d5 --- /dev/null +++ b/2020/round2/elimination.out @@ -0,0 +1,49701 @@ +Case #1: +1.00000000 +1.00000000 +Case #2: +1.33333333 +1.66666667 +2.00000000 +Case #3: +1.66666667 +1.94444444 +2.38888889 +3.00000000 +Case #4: +1.66666667 +1.66666667 +1.66666667 +Case #5: +2.47452857 +2.61417298 +2.77465280 +2.96047982 +3.17616583 +Case #6: +3.64469088 +3.83725603 +4.05825232 +4.31581280 +4.62193375 +4.99526004 +5.46668555 +6.09114388 +6.96896476 +Case #7: +1.00000000 +1.00000000 +Case #8: +8.39485659 +8.54168360 +8.69543966 +8.85672396 +9.02621582 +9.20468973 +9.39303409 +9.59227490 +9.80360586 +10.02842729 +10.26839692 +10.52549733 +10.80212673 +11.10122353 +11.42644014 +11.78239100 +12.17501397 +12.61211011 +13.10417060 +13.66567817 +Case #9: +67.04469643 +67.21298182 +67.38228630 +67.55262022 +67.72399406 +67.89641847 +68.06990424 +68.24446234 +68.42010389 +68.59684017 +68.77468265 +68.95364295 +69.13373288 +69.31496442 +69.49734974 +69.68090120 +69.86563133 +70.05155289 +70.23867879 +70.42702220 +70.61659644 +70.80741508 +70.99949190 +71.19284088 +71.38747624 +71.58341242 +71.78066411 +71.97924623 +72.17917393 +72.38046264 +72.58312802 +72.78718600 +72.99265277 +73.19954480 +73.40787883 +73.61767191 +73.82894134 +74.04170475 +74.25598006 +74.47178550 +74.68913964 +74.90806133 +75.12856981 +75.35068461 +75.57442566 +75.79981319 +76.02686785 +76.25561064 +76.48606293 +76.71824652 +76.95218357 +77.18789670 +77.42540890 +77.66474365 +77.90592483 +78.14897679 +78.39392436 +78.64079284 +78.88960801 +79.14039618 +79.39318414 +79.64799926 +79.90486941 +80.16382305 +80.42488919 +80.68809746 +80.95347807 +81.22106186 +81.49088031 +81.76296557 +82.03735045 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+9.86897136 +9.97822110 +10.09089139 +10.20718230 +10.32731187 +10.45151842 +10.58006309 +10.71323288 +10.85134414 +10.99474671 +11.14382876 +11.29902257 +11.46081136 +11.62973747 +11.80641228 +11.99152806 +12.18587255 +12.39034670 +12.60598659 +Case #109: +8.84808503 +8.85950339 +8.87096432 +8.88246812 +8.89401507 +8.90560547 +8.91723962 +8.92891782 +8.94064038 +8.95240760 +8.96421979 +8.97607728 +8.98798036 +8.99992938 +9.01192464 +9.02396648 +9.03605523 +Case #110: +4.51879116 +4.60032116 +4.68594211 +4.77601689 +4.87095427 +4.97121614 +5.07732600 +5.18987896 +5.30955332 diff --git a/2020/round2/elimination_sol.md b/2020/round2/elimination_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..90068c9b86154b40ec239df9e2352f8165f96da7 --- /dev/null +++ b/2020/round2/elimination_sol.md @@ -0,0 +1,8 @@ +This problem can be solved with dynamic programming. Let \(E_{b, w}\) be the expected number of future matches which a player will be present for given that they're currently in the tournament together with \(b\) better and \(w\) worse players than them. The \(i\)th output (the expected number of matches for which player \(i\) will be present) will then be \(E_{N - i, i - 1}\). + +Let's define \(F_{b, w}\) similarly to \(E_{b, w}\), except with the additional constraint that the winner of the next match is guaranteed to be the higher-skilled of the two competitors. We'll define \(G_{b, w}\) in much the same way, except with the next match's winner being the lower-skilled competitor. We then have \(E_{b, w} = F_{b, w} * P + G_{b, w} * (1 - P)\). + +We'll next consider how to compute \(F_{b, w}\), noting that \(G_{b, w}\) may then be computed very similarly. \(F_{b, w}\) is equal to \(1\) (as the player will certainly be present for the next match) plus the expected number of matches the player will be present after the next match (if they're not eliminated in it). If they're not about to be eliminated, then either a better or a worse player must be eliminated instead. Let \(Q(x) = \frac{x(x - 1)}{2}\) (the number of unordered pairs out of \(x\) players). If \(b > 0\), then there's a \(\frac{Q(b)}{Q(b + w + 1)}\) chance that a better player will be eliminated, meaning that we should add \(E_{b-1, w} * \frac{Q(b)}{Q(b + w + 1)}\) onto \(F_{b, w}\). Similarly, if \(w > 0\), we should add \(E_{b, w-1} * \frac{w(b+1) + Q(w)}{Q(b + w + 1)}\) onto \(F_{b, w}\). + +Over the course of the algorithm, we'll compute \(E_{b, w}\) for \(O(N^2)\) pairs of \(b\) and \(w\). Computing each such value takes constant time, resulting in a time complexity of \(O(N^2)\). + diff --git a/2020/round2/log_drivin_hirin.cpp b/2020/round2/log_drivin_hirin.cpp new file mode 100644 index 0000000000000000000000000000000000000000..8c03fe5976092cb75a6cc4ef9cf91b9a47f4eb03 --- /dev/null +++ b/2020/round2/log_drivin_hirin.cpp @@ -0,0 +1,190 @@ +// Log Drivin' Hirin' +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) { + char c, r = 0, n = 0; + x = 0; + for (;;) { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else if (r) + break; + } + if (n) + x = -x; + return(1); +} + +struct Line { + mutable LL k, m, p; + bool operator<(const Line& o) const { return k < o.k; } + bool operator<(LL x) const { return p < x; } +}; + +struct LineContainer : multiset> { + const LL inf = LLONG_MAX; + LL div(LL a, LL b) { // floored division + return a / b - ((a ^ b) < 0 && a % b); + } + bool isect(iterator x, iterator y) { + if (y == end()) { x->p = inf; return false; } + if (x->k == y->k) x->p = x->m > y->m ? inf : -inf; + else x->p = div(y->m - x->m, x->k - y->k); + return x->p >= y->p; + } + void add(LL k, LL m) { + auto z = insert({ k, m, 0 }), y = z++, x = y; + while (isect(y, z)) z = erase(z); + if (x != begin() && isect(--x, y)) isect(x, y = erase(y)); + while ((y = x) != begin() && (--x)->p >= y->p) + isect(x, erase(y)); + } + LL query(LL x) { + assert(!empty()); + auto l = *lower_bound(x); + return l.k * x + l.m; + } +}; + +#define LIM 1000001 +#define MOD 1000000007 + +int N, M, K, ans; +int P[LIM], L[LIM], H[LIM], X[LIM], Y[LIM]; +vector ch[LIM], Q[LIM]; +int nid, id[LIM]; +LineContainer LC[LIM]; + +void ReadSeq(int* V, int N, int K, int mod) +{ + int i, A, B, C; + Fox(i, K) + Read(V[i]); + Read(A), Read(B), Read(C); + if (!mod) + Read(mod); + FoxI(i, K, N - 1) + V[i] = ((LL)A * V[i - 2] + (LL)B * V[i - 1] + C) % (mod < 0 ? i : mod) + 1; +} + +LL GetQueryAns(int i, LL d, int c) +{ + return(c * d + LC[id[i]].query(c)); +} + +void rec(int i, LL d) +{ + // init new line container + id[i] = nid++; + // recurse to children + for (auto c : ch[i]) + { + rec(c, d + L[c]); + // merge smaller line container into larger one + int& a = id[i], & b = id[c]; + if (Sz(LC[a]) < Sz(LC[b])) + swap(a, b); + for (auto ln : LC[b]) + LC[a].add(ln.k, ln.m); + LC[b].clear(); + } + // add line for hireable log driver or leaf node + LC[id[i]].add(-d, Sz(ch[i]) ? GetQueryAns(i, d, H[i]) : 0); + // answer queries at this node + for (auto c : Q[i]) + ans = ans * ((-GetQueryAns(i, d, c) + 1) % MOD) % MOD; + ch[i].clear(), Q[i].clear(); +} + +int ProcessCase() +{ + int i; + // input + Read(N), Read(M), Read(K); + ReadSeq(P, N, K, -1); + ReadSeq(L, N, K, 0); + ReadSeq(H, N, K, 0); + ReadSeq(X, M, K, N); + ReadSeq(Y, M, K, 0); + // init tree/queries + Fox1(i, N - 1) + ch[P[i] - 1].pb(i); + Fox(i, M) + Q[X[i] - 1].pb(Y[i]); + // compute answers + ans = 1; + nid = 0; + rec(0, 0); + LC[id[0]].clear(); + return(ans); +} + +int main() { + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/round2/log_drivin_hirin.in b/2020/round2/log_drivin_hirin.in new file mode 100644 index 0000000000000000000000000000000000000000..15d493e6e496fee1d80952d1e4b383a56b1e1b9c --- /dev/null +++ b/2020/round2/log_drivin_hirin.in @@ -0,0 +1,1211 @@ +110 +3 4 3 +0 1 2 +0 0 0 +0 2 3 +0 0 0 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115636 9365 115635 9366 115634 9367 115633 9368 115632 9369 115631 9370 115630 9371 115629 9372 115628 9373 115627 9374 115626 9375 115625 9376 115624 9377 115623 9378 115622 9379 115621 9380 115620 9381 115619 9382 115618 9383 115617 9384 115616 9385 115615 9386 115614 9387 115613 9388 115612 9389 115611 9390 115610 9391 115609 9392 115608 9393 115607 9394 115606 9395 115605 9396 115604 9397 115603 9398 115602 9399 115601 9400 115600 9401 115599 9402 115598 9403 115597 9404 115596 9405 115595 9406 115594 9407 115593 9408 115592 9409 115591 9410 115590 9411 115589 9412 115588 9413 115587 9414 115586 9415 115585 9416 115584 9417 115583 9418 115582 9419 115581 9420 115580 9421 115579 9422 115578 9423 115577 9424 115576 9425 115575 9426 115574 9427 115573 9428 115572 9429 115571 9430 115570 9431 115569 9432 115568 9433 115567 9434 115566 9435 115565 9436 115564 9437 115563 9438 115562 9439 115561 9440 115560 9441 115559 9442 115558 9443 115557 9444 115556 9445 115555 9446 115554 9447 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114512 10489 114511 10490 114510 10491 114509 10492 114508 10493 114507 10494 114506 10495 114505 10496 114504 10497 114503 10498 114502 10499 114501 10500 114500 10501 114499 10502 114498 10503 114497 10504 114496 10505 114495 10506 114494 10507 114493 10508 114492 10509 114491 10510 114490 10511 114489 10512 114488 10513 114487 10514 114486 10515 114485 10516 114484 10517 114483 10518 114482 10519 114481 10520 114480 10521 114479 10522 114478 10523 114477 10524 114476 10525 114475 10526 114474 10527 114473 10528 114472 10529 114471 10530 114470 10531 114469 10532 114468 10533 114467 10534 114466 10535 114465 10536 114464 10537 114463 10538 114462 10539 114461 10540 114460 10541 114459 10542 114458 10543 114457 10544 114456 10545 114455 10546 114454 10547 114453 10548 114452 10549 114451 10550 114450 10551 114449 10552 114448 10553 114447 10554 114446 10555 114445 10556 114444 10557 114443 10558 114442 10559 114441 10560 114440 10561 114439 10562 114438 10563 114437 10564 114436 10565 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log drive. But as every Canadian knows, it can be a double-edged sword. If logs get stuck on the riverbanks, they'll eventually pile up and cause the river to overflow. [Sometimes you need a helping hand to keep the logs moving](https://youtu.be/7AheBqBk0iQ?t=578). + +In a given river system there are \(N\) lakes, numbered from \(1\) to \(N\), and \(N-1\) rivers flowing amongst the lakes. For each lake \(i\) such that \(2 \le i \le N\), there's a river flowing from lake \(P_i\) to lake \(i\) which is \(L_i\) metres long. There are no rivers flowing into lake 1. \(P_1\) and \(L_1\) are defined to be equal to 0, but are irrelevant. It's guaranteed that each lake is accessible from lake 1 by some sequence of rivers. Each lake with no rivers flowing out of it has a log-processing plant. + +Shipments of logs are often transported through these lakes and rivers. Each shipment consists of some number of logs, and is managed by a log driver with some integer carelessness value. A shipment may be transported from a lake by means of a river flowing out of it, to the lake which that river flows into. However, a number of logs will be permanently lost along the way, equal to the log driver's carelessness value multiplied by the length of the river (in metres). We can assume that shipments are always large enough that they don't risk completely running out of logs. Each shipment continues to be transported from lake to lake until reaching a log-processing plant. + +Other log drivers available for hire also live by the lakes. There's a log driver living by each lake \(i\) with a carelessness of \(H_i\). Whenever a shipment of logs is at lake \(i\), the company that owns the shipment may choose to hire the log driver there to take over in place of the shipment's current log driver. The newly-hired log driver will then manage the shipment until it either reaches a log-processing plant, or another log driver is later hired to replace them. + +As part of the log-istics team, you need to answer \(M\) questions, the \(i\)th of which asks: If you begin with a shipment of logs in lake \(X_i\), being managed by a log driver with a carelessness of \(Y_i\), what's the minimum number of logs which can be lost by the time the shipment reaches any log-processing plant? Please note that each question is independent of the others (for example, the same log driver may be hired as part of the answer to multiple different questions). + +Let \(V_i\) be the answer to the \(i\)th question. In order to reduce the size of the output, you must calculate a single integer: the product of \((V_i + 1)\) over \(i=1..M\). As this product may be very large, you should only compute its value modulo 1,000,000,007. + +In order to reduce the size of the input, the above values will not all be provided explicitly. Instead, you'll be given the first \(K\) values \(P_{1..K}\), \(L_{1..K}\), \(H_{1..K}\), \(X_{1..K}\), and \(Y_{1..K}\), as well as the five tuples of constants \((A_P, B_P, C_P)\), \((A_L, B_L, C_L, D_L)\), \((A_H, B_H, C_H, D_H)\), and \((A_X, B_X, C_X)\), \((A_Y, B_Y, C_Y, D_Y)\), and must then compute \(P_{(K+1)..N}\), \(L_{(K+1)..N}\), \(H_{(K+1)..N}\), \(X_{(K+1)..M}\), and \(Y_{(K+1)..M}\) as follows: + +\(P_i = ((A_P * P_{i-2} + B_P * P_{i-1} + C_P) \text{ modulo } (i-1)) + 1\) for \(i > K\) +\(L_i = ((A_L * L_{i-2} + B_L * L_{i-1} + C_L) \text{ modulo } D_L) + 1\) for \(i > K\) +\(H_i = ((A_H * H_{i-2} + B_H * H_{i-1} + C_H) \text{ modulo } D_H) + 1\) for \(i > K\) +\(X_i = ((A_X * X_{i-2} + B_X * X_{i-1} + C_X) \text{ modulo } N) + 1\) for \(i > K\) +\(Y_i = ((A_Y * Y_{i-2} + B_Y * Y_{i-1} + C_Y) \text{ modulo } D_Y) + 1\) for \(i > K\) + +# Constraints + +\(1 \le T \le 85\) +\(2 \le N, M \le 1,000,000\) +\(2 \le K \le min(N, M)\) +\(0 \le A_P, B_P, C_P, A_L, B_L, C_L, A_H, B_H, C_H, A_X, B_X, C_X, A_Y, B_Y, C_Y <= 1,000,000,000\) +\(1 \le D_L, D_H, D_Y \le 1,000,000\) +\(0 \le P_i \le i-1\) (\(P_i = 0\) if and only if \(i = 1\)) +\(0 \le L_i \le D_L\) (\(L_i = 0\) if and only if \(i = 1\)) +\(1 \le H_i \le D_H\) +\(1 \le X_i \le N\) +\(1 \le Y_i \le D_Y\) + +The sum of \(N+M\) over all \(T\) test cases is at most 10,000,000. + +# Input + +Input begins with an integer \(T\), the number of river systems. For each system, there are 11 lines: + +The first line contains the 3 space-separated integers \(N\), \(M\), and \(K\). + +The second line contains the \(K\) space-separated integers \(P_{1..K}\). + +The third line contains the 3 space-separated integers \(A_P\), \(B_P\), and \(C_P\). + +The fourth line contains the \(K\) space-separated integers \(L_{1..K}\). + +The fifth line contains the 4 space-separated integers \(A_L\), \(B_L\), \(C_L\), and \(D_L\). + +The sixth line contains the \(K\) space-separated integers \(H_{1..K}\). + +The seventh line contains the 4 space-separated integers \(A_H\), \(B_H\), \(C_H\), and \(D_H\). + +The eighth line contains the \(K\) space-separated integers \(X_{1..K}\). + +The ninth line contains the 3 space-separated integers \(A_X\), \(B_X\), and \(C_X\). + +The tenth line contains the \(K\) space-separated integers \(Y_{1..K}\). + +The eleventh line contains the 4 space-separated integers \(A_Y\), \(B_Y\), \(C_Y\), and \(D_Y\). + +# Output + +For the \(i\)th river system, print a line containing *"Case #i: "* followed by a single integer, the product of \((V_i + 1)\) over \(i=1..M\), modulo 1,000,000,007. + +# Explanation of Sample + +**In the first case:** + +- \(P = [0, 1, 2]\) +- \(L = [0, 2, 3]\) +- \(H = [10, 5, 50]\) +- \(X = [1, 1, 2, 3]\) +- \(Y = [2, 8, 3, 1]\) + +The system of lakes and rivers looks as follows (with hireable log drivers' carelessness values indicated in circles, and lakes with log-processing plants marked in brown): + +{{PHOTO_ID:896677524192910}} + +- The first question deals with a shipment of logs at lake \(1\) being managed by a log driver with carelessness \(2\). It's optimal for this log driver to move the shipment to lake \(2\) (while losing \(2 * 2 = 4\) logs), and then to lake \(3\) (losing another \(3 * 2 = 6\) logs), with a total of \(10\) logs lost. +- For the second question, the log driver should move the shipment to lake \(2\) and then be replaced by the log driver living there (who has a carelessness of \(5\)), who will move it to lake \(3\). This would result in a total of \(2 * 8 + 3 * 5 = 31\) logs lost. +- For the third question, the log driver should move the shipment to lake \(3\), losing \(3 * 3 = 9\) logs along the way. +- For the fourth question, the shipment begins at a lake with a log-processing plant, meaning that no logs need to be lost. + +Therefore, \(V = [10, 31, 9, 0]\), and the final answer is \(((10 + 1) * (31 + 1) * (9 + 1) * (0 + 1))\text{ modulo 1,000,000,007} = 3520\). + +**In the second case:** + +- \(P = [0, 1, 2, 1, 2, 4]\) +- \(L = [0, 11, 12, 13, 14, 15]\) +- \(H = [5, 4, 7, 2, 1, 3]\) +- \(X = [1, 2, 4, 1, 2, 1]\) +- \(Y = [1, 2, 3, 4, 5, 6]\) + +The system of lakes and rivers looks as follows: + +{{PHOTO_ID:3296189243803342}} + +For the first question, the log driver should move the shipment to lake \(2\) and then to lake \(3\), losing a total of \(11 * 1 + 12 * 1 = 23\) logs along the way. Overall, \(V = [23, 24, 30, 82, 48, 95]\). + +**In the third case:** + +- \(P = [0, 1, 2, 2, 2, 1, 6, 5]\) +- \(L = [0, 5, 1, 8, 3, 2, 9, 8]\) +- \(H = [10, 9, 8, 7, 6, 5, 4, 3]\) +- \(X = [5, 1, 2, 1, 8, 5, 4, 1, 8, 5]\) +- \(Y = [6, 2, 9, 8, 9, 4, 9, 4, 9, 4]\) + +For this case, \(V = [48, 12, 9, 48, 0, 32, 0, 24, 0, 32]\). diff --git a/2020/round2/log_drivin_hirin.out b/2020/round2/log_drivin_hirin.out new file mode 100644 index 0000000000000000000000000000000000000000..1872150ab9b6ed256549ee4927049d57a3ddebb1 --- /dev/null +++ b/2020/round2/log_drivin_hirin.out @@ -0,0 +1,110 @@ +Case #1: 3520 +Case #2: 262035151 +Case #3: 497739194 +Case #4: 762796014 +Case #5: 181526659 +Case #6: 2373746 +Case #7: 619271932 +Case #8: 223386728 +Case #9: 414974748 +Case #10: 976780998 +Case #11: 250999253 +Case #12: 408939742 +Case #13: 192239994 +Case #14: 296712448 +Case #15: 193611783 +Case #16: 587510705 +Case #17: 288849396 +Case #18: 270487397 +Case #19: 800731698 +Case #20: 816824727 +Case #21: 45287455 +Case #22: 283112753 +Case #23: 837057585 +Case #24: 515062771 +Case #25: 877386280 +Case #26: 714186675 +Case #27: 143891765 +Case #28: 708835696 +Case #29: 359065124 +Case #30: 781598349 +Case #31: 873170307 +Case #32: 897361282 +Case #33: 644903060 +Case #34: 864637537 +Case #35: 236700083 +Case #36: 705311549 +Case #37: 419732973 +Case #38: 84815421 +Case #39: 626168389 +Case #40: 503402652 +Case #41: 630535797 +Case #42: 74873913 +Case #43: 100018793 +Case #44: 807449687 +Case #45: 373341778 +Case #46: 205510051 +Case #47: 56084597 +Case #48: 435911090 +Case #49: 222481170 +Case #50: 443116000 +Case #51: 374749071 +Case #52: 856977477 +Case #53: 542120123 +Case #54: 838789422 +Case #55: 747793126 +Case #56: 701291130 +Case #57: 291535215 +Case #58: 612509140 +Case #59: 785302203 +Case #60: 927126582 +Case #61: 449828522 +Case #62: 41974773 +Case #63: 452621798 +Case #64: 503016535 +Case #65: 572021988 +Case #66: 433100162 +Case #67: 723846663 +Case #68: 682235539 +Case #69: 956070458 +Case #70: 82898978 +Case #71: 647943426 +Case #72: 655213032 +Case #73: 619570132 +Case #74: 81954741 +Case #75: 33426022 +Case #76: 441677902 +Case #77: 1366253 +Case #78: 401758825 +Case #79: 566900 +Case #80: 951034217 +Case #81: 898434034 +Case #82: 259753767 +Case #83: 267376664 +Case #84: 648667847 +Case #85: 755395859 +Case #86: 495622006 +Case #87: 617580551 +Case #88: 452172976 +Case #89: 810298524 +Case #90: 158301234 +Case #91: 236762888 +Case #92: 996262703 +Case #93: 180994338 +Case #94: 308450384 +Case #95: 243262686 +Case #96: 765758595 +Case #97: 564061278 +Case #98: 306236296 +Case #99: 542733267 +Case #100: 302156108 +Case #101: 990333199 +Case #102: 779042221 +Case #103: 367335896 +Case #104: 121781160 +Case #105: 275197625 +Case #106: 477751182 +Case #107: 956233494 +Case #108: 406689005 +Case #109: 511969556 +Case #110: 378685288 diff --git a/2020/round2/log_drivin_hirin_sol.md b/2020/round2/log_drivin_hirin_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..2c405795cbd46243538f8971e5cb875112769a04 --- /dev/null +++ b/2020/round2/log_drivin_hirin_sol.md @@ -0,0 +1,14 @@ +The lakes and rivers may be represented as a tree rooted at node (lake) \(1\). Let \(D_i\) be the total distance along rivers from the root to each node \(i\). + +Let \(Q(i, c)\) be the minimum number of logs which may be lost for a shipment of logs which begins at node \(i\) while being managed by a log driver with a carelessness of \(c\) (such that \(V_i = Q(X_i, Y_i)\)). + +If node \(i\) has a log-processing plant (i.e. is a leaf node), \(Q(i, c) = 0\) for any \(c\). Otherwise, \(Q(i, c)\) may be computed by considering possible options for the first "relevant node" \(j\) encountered by a shipment beginning at node \(i\), such that either \(j\) is a leaf node or the first node at which a new log driver will be hired (note that \(j\) must be within \(i\)'s subtree, and may be equal to \(i\) itself). For a given choice of \(j\), the number of logs lost for the shipment comes out to \(c(D_j - D_i) + Q(j, H_j)\), and so we're looking for the node \(j\) which minimizes this expression. + +Rewriting the expression as \(cD_j + Q(j, H_j) - cD_i\), and ignoring the final portion (which is independent of \(j\)), we can observe that it forms a linear equation \(F_j(c) = m_j c + b_j\) (where \(m_j = D_j\) and \(b_j = Q(j, H_j)\)). As such, we've reduced computing \(Q(i, c)\) to finding the minimum value of \(F_j(c)\) over all nodes \(j\) in \(i\)'s subtree. To avoid an infinitely recursive definition, note that we can evaluate \(Q(i, H_i)\) by only considering nodes \(j\) such that \(j \ne i\). + +Our approach will involve recursively traversing the tree, such that for each node \(i\), we compute \(Q(i, H_i)\) followed by \(Q(X_k, Y_k)\) for each question such that \(X_k = i\). Along the way, we'll need to maintain information around the linear equations \(F_j(c)\) for all nodes \(j\) in \(i\)'s subtree, so that we can efficiently compute each \(Q(i, c)\) value. + +Making use of the *dynamic convex hull trick*, we'll store these linear equations in a set ordered by their slope (\(m_y\)), such that each equation yields the minimum result out of all equations when evaluated for some non-empty interval of \(c\) values. Our data structure will also store the "intersection points" of all consecutive pairs of linear equations in the ordered set, which correspond to the intervals of \(c\) values for which the various equations are optimal. With this setup, we're able to insert a new equation into the set and to query the minimum value of any equation for a given value of \(c\) in \(O(log(n))\) time each (where \(n\) is the number of equations in the set). + +The remaining question is how we'll build up these sets of equations for entire subtrees of nodes. When handling each node \(i\), we must efficiently merge all of its children's sets together, before proceeding to compute and insert \(F_i\) as well. When merging two sets together, we should simply elect to reuse the larger of the two (or an arbitrary one if they have the same size), inserting all equations from the smaller one into it and then discarding that smaller set. Each time a given equation is merged into a different set, the size of its containing set at least doubles, meaning that it may be inserted into sets at most \(O(log(N))\) times. With each set consisting of up to \(N\) equations, the total time complexity of all insertions comes out to \(O(N log^2(N))\). Factoring in the set queries required to evaluate all \(N + M\) needed \(Q(i, c)\) values, we arrive at a total of \(O(N log^2(N) + M log(N))\) time. + diff --git a/2020/round3/1217734278611436.jpg b/2020/round3/1217734278611436.jpg new file mode 100644 index 0000000000000000000000000000000000000000..8251d65b6f4c0ba2fc5e04c92a42881a60ad1884 --- /dev/null +++ b/2020/round3/1217734278611436.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6338cdfe308703c1e789fb483c20c9dfe905ea220a74c098edccfa3e8f79755d +size 5206 diff --git a/2020/round3/2943967372557617.jpg b/2020/round3/2943967372557617.jpg new file mode 100644 index 0000000000000000000000000000000000000000..1262b9b4f375eff1b964054e09d1a131f0c1625a --- /dev/null +++ b/2020/round3/2943967372557617.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:43ac940b689dd6a22d1dbdfc7b92a928556ffe83be85ce4a9b7ab826fda61ffc +size 12607 diff --git a/2020/round3/2978496958918891.jpg b/2020/round3/2978496958918891.jpg new file mode 100644 index 0000000000000000000000000000000000000000..bd7b58e3e4593f403348c70ed5f666ca9b9820b9 --- /dev/null +++ b/2020/round3/2978496958918891.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5fbb183c464e1b0b347a615779893d7dae447721633a84ae740f63b08c9168d0 +size 6613 diff --git a/2020/round3/326196371830360.jpg b/2020/round3/326196371830360.jpg new file mode 100644 index 0000000000000000000000000000000000000000..528cf6e0c617c389c65cd2a8931523fa1f99e4c4 --- /dev/null +++ b/2020/round3/326196371830360.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:203832554801edcac6f23bc36a29347ce870d336937b04f6017a3582b48f5a0c +size 20253 diff --git a/2020/round3/3274155842706395.jpg b/2020/round3/3274155842706395.jpg new file mode 100644 index 0000000000000000000000000000000000000000..28a026d734a139e51fb692412fb51bd9cf1ff189 --- /dev/null +++ b/2020/round3/3274155842706395.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ee3112b759a5fc02299e3eb1a7c877b45eb4040cb2003a53a9e06d3a023a57a3 +size 7627 diff --git a/2020/round3/3302289739857297.jpg b/2020/round3/3302289739857297.jpg new file mode 100644 index 0000000000000000000000000000000000000000..0454445d019e172eb0bb7456fdcc70622c3687e5 --- /dev/null +++ b/2020/round3/3302289739857297.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8a2626d8995f09246a05407b8128d51f6c6f2549b3f04382e09ccf6889952dcb +size 5550 diff --git a/2020/round3/3371452566235882.jpg b/2020/round3/3371452566235882.jpg new file mode 100644 index 0000000000000000000000000000000000000000..3175561dcacab181cf4acc12a4ed6ac722a2253e --- /dev/null +++ b/2020/round3/3371452566235882.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:891d7e1fceae6c097848c63904ff4b5a2f4a1079da968aeadd66e4f4190ba022 +size 2914 diff --git a/2020/round3/636021084002078.jpg b/2020/round3/636021084002078.jpg new file mode 100644 index 0000000000000000000000000000000000000000..0a0582ba3594a9a9de28d77b259919345d20041b --- /dev/null +++ b/2020/round3/636021084002078.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:25de6792372585f0aa0d406a2c7d636049ffcc6a90b155597e44cf9672a9acc4 +size 9190 diff --git a/2020/round3/766911834140516.jpg b/2020/round3/766911834140516.jpg new file mode 100644 index 0000000000000000000000000000000000000000..df413401cdbddd62d40d19893ddacb174e62dcf7 --- /dev/null +++ b/2020/round3/766911834140516.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9f8ecfc5b1282acebd591d5dfca06c15093a0eaf884da7ae4603dd934a7f7159 +size 5696 diff --git a/2020/round3/chain_explosions.cpp b/2020/round3/chain_explosions.cpp new file mode 100644 index 0000000000000000000000000000000000000000..1ee7641adc9c803ff4c837c3c8d63a91245f1312 --- /dev/null +++ b/2020/round3/chain_explosions.cpp @@ -0,0 +1,113 @@ +// Chain Explosions +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +void ProcessCase() +{ + // input + int K; + Read(K), K /= 2; + // construct graph + int N = 2; + vector E = { { 1, 2 } }; + while (K) + { + int r = N, d = 1; + while (d <= K) + E.pb({ r, ++N }), K -= d++; + } + // output + printf("%d %d\n", N, Sz(E)); + for (auto e : E) + printf("%d %d\n", e.x, e.y); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + { + printf("Case #%d: ", t); + ProcessCase(); + } + return(0); +} \ No newline at end of file diff --git a/2020/round3/chain_explosions.in b/2020/round3/chain_explosions.in new file mode 100644 index 0000000000000000000000000000000000000000..ff5f8f430b0405e256cd7c716b9016f890cb8b36 --- /dev/null +++ b/2020/round3/chain_explosions.in @@ -0,0 +1,103 @@ +102 +2 +6 +18 +4 +10 +20 +100 +100000 +8 +12 +14 +16 +22 +24 +26 +28 +30 +32 +34 +36 +38 +40 +2000000 +1999998 +1987654 +1234566 +1997902 +1868 +1802 +1780 +1048 +1174 +1138 +282 +784 +716 +1174 +1292 +1550 +772 +1974 +530 +302 +130 +550 +1106 +420 +408 +1354 +1944 +1622 +540 +1644 +1488 +80 +1010 +1990 +674 +1840 +400 +1452 +1976 +1382 +678 +1578 +538 +1488 +1330 +654 +1370 +1534 +384 +264 +128 +248 +1312 +864 +260 +824 +250 +1926 +1810 +1488 +726 +800 +1468 +278 +232 +160 +1338 +722 +1610 +1236 +40 +1804 +892 +20 +642 +14 +624 +358 +600 diff --git a/2020/round3/chain_explosions.md b/2020/round3/chain_explosions.md new file mode 100644 index 0000000000000000000000000000000000000000..10c37326e4ac1d0a6b235f4655f610c577c26f96 --- /dev/null +++ b/2020/round3/chain_explosions.md @@ -0,0 +1,63 @@ +Dynamite is an [effective tool](https://youtu.be/7AheBqBk0iQ?t=433) for clearing ice to allow logs to flow downstream in the great Canadian log drive. But as every Canadian knows, successful use of dynamite requires a certain finesse. Logan the logging director is planning to hook up a network of dynamite sticks using fuse wires so that they may be detonated from afar. + +Logan would like to use at most 2000 sticks of dynamite and at most 2000 fuse wires to build his dynamite network at a certain job site. He'll number the sticks from \(1\) to \(N\), and the wires from \(1\) to \(M\) (for some \(N\) and \(M\) of his choice). Each fuse wire is used to connect two different sticks of dynamite together. No two fuse wires may be used to connect the same unordered pair of sticks. The resulting dynamite network must all be connected (that is, it must be possible to reach each dynamite stick from each other stick by following a sequence of connected wires and sticks). + +There's just one problem: sticks of dynamite are not individually powerful enough to break the thick ice. Logan must rely on *chain explosions* to amplify the dynamite's power. A potential chain explosion can occur anywhere in Logan's network where there is an ordered triple \((a, b, c)\) of distinct sticks of dynamite such that sticks \(a\) and \(b\) are connected by a fuse wire, as are sticks \(b\) and \(c\). Logan would like to build a network with exactly \(K\) potential chain explosions, where \(K\) is an even integer. Any less than \(K\), and there might not be enough ice-breaking power; any more than \(K\), and he risks damaging the logs. + +Help Logan bust some ice by constructing *any* valid dynamite network for the given value of \(K\). Please note that at least one valid network exists for each possible valid value of \(K\). + +# Constraints + +\(1 \le T \le 100\) +\(2 \le K \le 2,000,000\) +\(K\) is even + +The sum of \(K\) across all job sites is at most 10,000,000. + +# Input + +Input begins with an integer \(T\), the number of job sites. For each job site, there is a line containing a single integer \(K\). + +# Output + +For the \(i\)th job site, print a line containing *"Case #i: "*, followed by 2 integers \(N\) and \(M\), the number of sticks of dynamite and fuse wires your network will use \((1 \le N, M \le 2000)\). After that, print \(M\) more lines, the \(j\)th of which contains 2 integers \(A_j\) and \(B_j\), the two sticks of dynamite connected by the \(j\)th fuse wire \((1 \le A_j, B_j \le N, A_j \ne B_j)\). + + + +# Explanation of Sample + +***Note: For each job site, other outputs would also be accepted.*** + +For the first job site, the chosen network (depicted below) has exactly 2 potential chain explosions (involving dynamic stick triples \((1, 2, 3)\) and \((3, 2, 1)\)), as required. + +{{PHOTO_ID:636021084002078}} + +For the second job site, the chosen network (depicted below) has exactly 6 potential chain explosions (involving sticks \((1, 2, 3)\), \((1, 3, 2)\), \((2, 1, 3)\), \((2, 3, 1)\), \((3, 1, 2)\), and \((3, 2, 1)\)). + +{{PHOTO_ID:2943967372557617}} + +The following output would *not* be accepted, due to having a fuse wire connecting a stick of dynamite to itself (stick 3) and having two fuse wires connecting the same unordered pair of dynamite sticks (sticks 1 and 2): + +``` +3 5 +1 2 +2 1 +2 3 +3 3 +3 1 +``` + +The following output would also *not* be accepted, due to the dynamite network not being connected: + +``` +7 5 +1 2 +2 3 +3 4 +5 6 +6 7 +``` + +For the third job site, the chosen network is as follows: + +{{PHOTO_ID:326196371830360}} diff --git a/2020/round3/chain_explosions.out b/2020/round3/chain_explosions.out new file mode 100644 index 0000000000000000000000000000000000000000..9fe25eac1185ed3b0db6807d5a6fec4d86a244fd --- /dev/null +++ b/2020/round3/chain_explosions.out @@ -0,0 +1,10334 @@ +Case #1: 3 2 +1 2 +1 3 +Case #2: 4 3 +1 2 +1 3 +1 4 +Case #3: 8 7 +1 2 +1 3 +1 4 +1 5 +5 7 +6 7 +6 8 +Case #4: 4 3 +1 2 +1 3 +3 4 +Case #5: 6 5 +1 2 +1 3 +1 4 +4 6 +5 6 +Case #6: 6 5 +1 2 +1 3 +1 4 +1 5 +1 6 +Case #7: 15 14 +1 2 +1 3 +1 4 +1 5 +1 6 +1 7 +1 8 +1 9 +1 10 +1 11 +11 13 +12 13 +12 14 +12 15 +Case #8: 347 346 +1 2 +1 3 +1 4 +1 5 +1 6 +1 7 +1 8 +1 9 +1 10 +1 11 +1 12 +1 13 +1 14 +1 15 +1 16 +1 17 +1 18 +1 19 +1 20 +1 21 +1 22 +1 23 +1 24 +1 25 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+1 17 +1 18 +1 19 +1 20 +1 21 +1 22 +1 23 +1 24 +1 25 +1 26 diff --git a/2020/round3/chain_explosions_sol.md b/2020/round3/chain_explosions_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..efc124f3fab5313a128fb523ca19f40e25044e5f --- /dev/null +++ b/2020/round3/chain_explosions_sol.md @@ -0,0 +1,9 @@ +We need to construct an undirected, connected graph with exactly \(K\) length-2 simple paths. The bounds indicate that we may only use up to roughly \(\sqrt{K}\) nodes and edges in this graph. + +If we let \(D_i\) be the degree of node \(i\), we can observe that there are \(D_i (D_i - 1)\) length-2 simple paths for which node \(i\) is the middle node. There are multiple ways to use this observation to construct a valid graph, with one simple method described below. + +We'll begin with a graph having 2 nodes (numbered \(1\) and \(2\)) connected by an edge. Note that this graph contains no length-2 simple paths. We'll then repeat the following process until it has \(K\) length-2 simple paths. + +Let \(r\) be the largest-numbered node currently present in the graph. We'll be constructing the graph such that the degree of the largest-numbered node at any point is always known to equal \(1\) (note that this is the case in the initial graph). We'll now repeatedly add new nodes to the graph (numbered \(r+1\), \(r+2\), and so on), connecting each one to node \(r\) by an edge, until adding another such node would cause the graph to have more than \(K\) length-2 simple paths. Each time such a node is added, \(2 D_r\) length-2 simple paths are added to the graph, and then \(D_r\) increases by \(1\). Note that each new node's degree is \(1\), and the last of the added nodes will become node \(r\) for the next iteration of the process (if any). + +The above approach uses fewer than 1,500 nodes and edges for any even \(K\) up to 2,000,000, comfortably under the limit of 2,000. diff --git a/2020/round3/mail_security.cpp b/2020/round3/mail_security.cpp new file mode 100644 index 0000000000000000000000000000000000000000..7199d5e39b94be0d3e68fe3690cea1b0c3c9aaf0 --- /dev/null +++ b/2020/round3/mail_security.cpp @@ -0,0 +1,164 @@ +// Mail Security +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 1000000 + +int N, M, K, X; +int S[LIM], P[LIM]; + +void ReadSeq(int* V, int N, int K) +{ + int i, A, B, C, D; + Fox(i, K) + Read(V[i]); + Read(A), Read(B), Read(C), Read(D); + FoxI(i, K, N - 1) + V[i] = ((LL)A * V[i - 2] + (LL)B * V[i - 1] + C) % D + 1; +} + +bool Valid(int np) +{ + int i; + // assemble list of relevant objects + vector O; + FoxR(i, N) // largest np mailboxes, plus any large enough for a key + if (O.size() < np || S[i] >= X) + O.pb(mp(S[i], 1)); + int nb = Sz(O); + Fox(i, np) // smallest np packages + O.pb(mp(P[i], 0)); + sort(All(O), std::greater<>()); + // iterate over objects by non-increasing capacity/size + multiset AC; // available pairs of (capacity % X, capacity) + LL nk = 0; + for (auto o : O) + { + // mailbox? + if (o.y) + { + nk += o.x / X; + AC.insert(mp(o.x % X, o.x)); + continue; + } + // package + if (AC.empty()) + return(0); + auto I = AC.lower_bound(mp(o.x % X, 0)); + if (I == AC.end()) + I = AC.begin(); + int c = I->y; + AC.erase(I); + nk += (c - o.x) / X - c / X; + } + return(nk >= nb - 1); +} + +int ProcessCase() +{ + // input + Read(N), Read(M), Read(K), Read(X); + ReadSeq(S, N, K); + ReadSeq(P, M, K); + // sort objects by capacity/size + sort(S, S + N); + sort(P, P + M); + // binary search + int r1 = 0, r2 = min(N, M); + while (r1 < r2) + { + int m = (r1 + r2 + 1) / 2; + if (Valid(m)) + r1 = m; + else + r2 = m - 1; + } + return(r1); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/round3/mail_security.in b/2020/round3/mail_security.in new file mode 100644 index 0000000000000000000000000000000000000000..1de9821d6a8ffdd674d47f42e87159ee63ddaf13 --- /dev/null +++ b/2020/round3/mail_security.in @@ -0,0 +1,566 @@ +113 +3 3 3 1 +10 10 10 +0 0 0 100 +9 9 9 +0 0 0 100 +3 3 3 1 +10 10 10 +0 0 0 100 +10 10 10 +0 0 0 100 +3 3 3 1 +10 10 10 +0 0 0 100 +11 11 11 +0 0 0 100 +4 4 2 3 +2 11 +1 0 6 11 +10 9 +0 1 0 13 +7 5 5 4 +15 3 5 11 7 +13 6 11 20 +7 8 11 10 3 +0 0 0 100 +20 25 2 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a/2020/round3/mail_security.md b/2020/round3/mail_security.md new file mode 100644 index 0000000000000000000000000000000000000000..0b371140d7af2e1c430c08d39c9ece9aebd11b6f --- /dev/null +++ b/2020/round3/mail_security.md @@ -0,0 +1,67 @@ +Facebook Prime is a top-secret subscription service being developed to disrupt the online shopping space. (Or maybe there's no such service — we're not telling.) A central focus of the service is preventing package theft, something which has become all too common in today's delivery-driven world. As a member of its engineering team, you wish to model some worst-case scenarios to stress test the vulnerability of poorly-run mailrooms. + +In the *very* worst cases, mailroom managers will actually leave mailbox keys inside mailboxes themselves. This means that if one mailbox were to be broken into, a series of other mailboxes could be compromised as well. Suppose a particular mailroom has \(N\) mailboxes (numbered from \(1\) to \(N\)), each with a corresponding key, and \(M\) packages (numbered from \(1\) to \(M\)). The \(i\)th mailbox has a capacity of \(S_i\), meaning that it can store a set of objects (packages or keys) as long as the sum of their sizes doesn't exceed \(S_i\). The size of the \(i\)th package is \(P_i\), and the size of each key is \(X\). + +For the purposes of our model, a valid mailroom arrangement must satisfy all of the following conditions: + + 1. Each package is either inside a mailbox, or is entirely absent. + 2. Each key is either inside a mailbox (possibly its own corresponding mailbox), or is entirely absent. + 3. Each mailbox contains at most 1 package. + 4. No mailbox's capacity is exceeded. + +Suppose a thief enters a validly arranged mailroom and forcibly breaks into just one mailbox (without needing a key). If the mailbox contains a package, then they'll obtain it; if the mailbox contains any keys, the thief can then proceed to open each of those keys' corresponding mailboxes and repeat the process (gaining access to any packages and keys they contain, and so on). + +Given the specifications of a mailroom, you wish to determine, among all possible valid arrangements, the largest number of packages that may be obtained after a thief forcibly breaks into a single mailbox. + +In order to reduce the size of the input, the above values will not all be provided explicitly. Instead, you'll be given the first K values \(S_{1..K}\) and \(P_{1..K}\), as well as the two quadruples of constants \((A_S, B_S, C_S, D_S)\) and \((A_P, B_P, C_P, D_P)\), and must then compute \(S_{(K+1)..N}\) and \(P_{(K+1)..M}\) as follows: + +\(S_i = ((A_S * S_{i-2} + B_S * S_{i-1} + C_S)\text{ modulo }D_S) + 1\) for \(i > K\) +\(P_i = ((A_P * P_{i-2} + B_P * P_{i-1} + C_P)\text{ modulo }D_P) + 1\) for \(i > K\) + +# Constraints + +\(1 \le T \le 90\) +\(2 \le N \le 1,000,000\) +\(2 \le M \le 1,000,000\) +\(2 \le K \le min(N, M)\) +\(1 \le X \le 1,000,000,000\) +\(0 \le A_S, B_S, C_S, A_P, B_P, C_P \le 1,000,000,000\) +\(1 \le D_S, D_P \le 1,000,000,000\) +\(1 \le S_i \le D_S\) +\(1 \le P_i \le D_P\) + +The sum of \(N + M\) across all mailrooms is at most 10,000,000. + +# Input + +Input begins with an integer \(T\), the number of mailrooms to be modeled. For each mailroom, there are 5 lines: + +The first line contains the 4 space-separated integers \(N\), \(M\), \(K\), and \(X\). + +The second line contains the \(K\) space-separated integers \(S_{1..K}\). + +The third line contains the 4 space-separated integers \(A_S\), \(B_S\), \(C_S\), and \(D_S\). + +The fourth line contains the \(K\) space-separated integers \(P_{1..K}\). + +The fifth line contains the 4 space-separated integers \(A_P\), \(B_P\), \(C_P\), and \(D_P\). + + +# Output + +For each mailroom, print a line containing *"Case #i: "*, followed by a single integer, the maximum possible number of packages which could be stolen by forcibly breaking into a single mailbox in any valid mailroom arrangement. + + + + +# Explanation of Sample + +For the first mailroom, \(S = [10, 10, 10]\) and \(P = [9, 9, 9]\). If each mailbox contains a package, the 1st mailbox contains the 2nd mailbox's key, and the 2nd mailbox contains the 3rd mailbox's key, then a thief breaking into the 1st mailbox would gain access to the 2nd mailbox followed by the 3rd, obtaining all three packages in the process. + +For the second mailroom, \(S = [10, 10, 10]\) and \(P = [10, 10, 10]\). If the 1st and 3rd mailboxes each contain a package, while the 2nd mailbox contains all three mailboxes' keys, then a thief breaking into the 2nd mailbox would gain access to the other two mailboxes and their packages. + +For the third mailroom, \(S = [10, 10, 10]\) and \(P = [11, 11, 11]\). No package can fit within any mailbox. + +For the fourth mailroom, \(S = [2, 11, 9, 7]\) and \(P = [10, 9, 10, 11]\). It's possible to arrange the mailroom such that two packages are obtainable by a thief, but three or more can never be obtained. + +For the fifth mailroom, \(S = [15, 3, 5, 11, 7, 17, 5]\) and \(P = [7, 8, 11, 10, 3]\). diff --git a/2020/round3/mail_security.out b/2020/round3/mail_security.out new file mode 100644 index 0000000000000000000000000000000000000000..31ce36d425ba383cb7908f01e2db02ae723e88e4 --- /dev/null +++ b/2020/round3/mail_security.out @@ -0,0 +1,113 @@ +Case #1: 3 +Case #2: 2 +Case #3: 0 +Case #4: 2 +Case #5: 4 +Case #6: 16 +Case #7: 0 +Case #8: 41 +Case #9: 3994 +Case #10: 4577 +Case #11: 2 +Case #12: 3 +Case #13: 2 +Case #14: 899501 +Case #15: 1000000 +Case #16: 520578 +Case #17: 63351 +Case #18: 2090 +Case #19: 509 +Case #20: 5226 +Case #21: 5484 +Case #22: 1036 +Case #23: 0 +Case #24: 360 +Case #25: 749 +Case #26: 2055 +Case #27: 2636 +Case #28: 4249 +Case #29: 130 +Case #30: 33 +Case #31: 3515 +Case #32: 192 +Case #33: 4620 +Case #34: 747 +Case #35: 911 +Case #36: 7288 +Case #37: 1 +Case #38: 4172 +Case #39: 1608 +Case #40: 111 +Case #41: 3410 +Case #42: 2604 +Case #43: 2071 +Case #44: 220 +Case #45: 379 +Case #46: 1 +Case #47: 1057 +Case #48: 305 +Case #49: 254 +Case #50: 0 +Case #51: 2428 +Case #52: 1 +Case #53: 318 +Case #54: 2570 +Case #55: 1 +Case #56: 2884 +Case #57: 1233 +Case #58: 338 +Case #59: 1001 +Case #60: 548 +Case #61: 6513 +Case #62: 2139 +Case #63: 909 +Case #64: 4948 +Case #65: 129 +Case #66: 810 +Case #67: 172 +Case #68: 967 +Case #69: 1364 +Case #70: 1 +Case #71: 1539 +Case #72: 1171 +Case #73: 1723 +Case #74: 2255 +Case #75: 9 +Case #76: 6762 +Case #77: 2985 +Case #78: 676 +Case #79: 2253 +Case #80: 2244 +Case #81: 2694 +Case #82: 1970 +Case #83: 1229 +Case #84: 1 +Case #85: 139 +Case #86: 2910 +Case #87: 1066 +Case #88: 50 +Case #89: 6508 +Case #90: 4068 +Case #91: 1296 +Case #92: 932 +Case #93: 1 +Case #94: 297 +Case #95: 193 +Case #96: 3305 +Case #97: 3519 +Case #98: 936 +Case #99: 1 +Case #100: 1 +Case #101: 6030 +Case #102: 1679 +Case #103: 2683 +Case #104: 3880 +Case #105: 1141 +Case #106: 331 +Case #107: 4040 +Case #108: 488 +Case #109: 1652 +Case #110: 1036 +Case #111: 2337 +Case #112: 2771 +Case #113: 4170 diff --git a/2020/round3/mail_security_sol.md b/2020/round3/mail_security_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..612f0e07852088064e370787d1c8dfe7822a3d23 --- /dev/null +++ b/2020/round3/mail_security_sol.md @@ -0,0 +1,14 @@ +As a general observation, we can note that keys are essentially interchangeable. If we're "using" \(x\) mailboxes (that is, there are \(x\) non-empty mailboxes that we intend for the thief to gain access to), the only requirement is that a total of at least \(x-1\) keys are present somewhere amongst them. It's then always possible to assign the keys' identities and select an initial break-in mailbox such that the thief will eventually access all \(x\) mailboxes. + +Now, we'll begin by sorting the mailboxes in order of capacity, and the packages in order of size. We'll then binary search on \([0, \min(N, M)]\) for the maximum number \(v\) of packages which can be obtained, which will be our answer. + +For a given value of \(v\), we'll begin by reducing the set of \(N+M\) mailboxes/packages to only the relevant ones which we'll intend to use. We should attempt to use the smallest \(v\) packages, ignoring the rest. We'll need to use at least \(v\) mailboxes, which might as well be the largest \(v\) of them. If there are additional mailboxes capable of fitting at least one key (having capacities no less than \(X\)), then we should intend on using all of them as well (even if some may prove unnecessary), and we should **ignore any remaining mailboxes**. It can't be better to use fewer mailboxes, as using a mailbox capable of fitting a key can never be detrimental (as it can at least fit a key allowing use of another mailbox in its place). It also can't be better to use more mailboxes, as using an *additional* mailbox with capacity less than \(X\) (with the intent of placing a package with size less than \(X\) in it) cannot be more beneficial than instead replacing an existing key in some larger mailbox not yet containing any package with this small package — at least one such mailbox must exist in that scenario. + +We'll then combine the relevant mailboxes/packages into a single list sorted in non-increasing order of capacity/size, and iterate over them in that order. Along the way, we'll maintain an ordered multiset of mailbox capacities which have been seen but not yet used, ordered by their capacities modulo \(X\) (initially empty), as well as the total number \(k\) of keys which can fit within the already-seen mailboxes (initially \(0\)). + +- When a mailbox with capacity \(c\) is encountered, we'll insert \(c\) into our multiset and increase \(k\) by \(\lfloor c/X \rfloor\). +- When a package with size \(p\) is encountered, it must be placed into some mailbox in our multiset. If the multiset is empty, then the given value of \(v\) must be invalid. Otherwise, we should choose a mailbox capacity \(c\) for which \(c \text{ \% } X \ge p \text{ \% } X\) and \(c \text{ \% } X\) is minimized — or, if there's no such capacity, then instead just a capacity \(c\) for which \(c \text{ \% } X\) is minimized (where \(\text{\%}\) is the modulo operator). This greedy strategy is based on the fact that the package will reduce the number of keys which can fit in the mailbox by \(\lfloor p/X \rfloor\) if \(c \text{ \% } X \ge p \text{ \% } X\), and by \(\lfloor p/X \rfloor + 1\) otherwise. Either way, once the optimal \(c\) has been selected, we'll remove it from our multiset and decrease \(k\) accordingly. + +By the end of this process, the given value of \(v\) must be valid if and only if \(k+1\) is greater than or equal to the number of mailboxes used. + +The total time complexity of this solution is \(O((N + M) (log(N) + log(M))^2)\). diff --git a/2020/round3/railroad_renovations.cpp b/2020/round3/railroad_renovations.cpp new file mode 100644 index 0000000000000000000000000000000000000000..0e3bf6d5ed6fde746c86e8f8d43594c271c2769d --- /dev/null +++ b/2020/round3/railroad_renovations.cpp @@ -0,0 +1,150 @@ +// Railroad Renovations +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 501 + +int N, K; +pair O[LIM]; + +// dyn[i][j] = min. renovations after first i observations (by position), +// with j rearrangements +int dyn[LIM][LIM]; + +int ProcessCase() +{ + int i, j, k; + // input + Read(N), Read(K); + Fox(i, N) + { + Read(O[i].x), Read(O[i].y.y); + O[i].y.x = i; + } + // sort observations by position + sort(O, O + N); + // DP + Fill(dyn, 60); + dyn[0][0] = 0; + Fox(i, N) + { + // can start interval here? + if (i - 1 >= 0 && O[i].x == O[i - 1].x) + continue; + // consider all intervals starting here + int C[2] = { 0 }; + set S; + FoxI(j, i, N - 1) + { + // update interval info + S.insert(O[j].y); + C[O[j].y.y]++; + // can end interval here? + if (j + 1 < N && O[j].x == O[j + 1].x) + continue; + // compute min. cost for this interval (all 0s before all 1s) + int cst = min(C[0], C[1]); + int pc[2] = { 0 }; + for (auto o : S) // consider all 0-1 splitting points + { + pc[o.y]++; + Min(cst, pc[1] + C[0] - pc[0]); + } + // update DP values + FoxI(k, cst, K) + Min(dyn[j + 1][k], dyn[i][k - cst] + (C[1] > 0)); + } + } + // compute answer + int ans = INF; + Fox(k, K + 1) + Min(ans, dyn[N][k]); + return(ans == INF ? -1 : ans); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + printf("Case #%d: %d\n", t, ProcessCase()); + return(0); +} \ No newline at end of file diff --git a/2020/round3/railroad_renovations.in b/2020/round3/railroad_renovations.in new file mode 100644 index 0000000000000000000000000000000000000000..059418fd82234c3dc92a0f521f97097e4ef8de92 --- /dev/null +++ b/2020/round3/railroad_renovations.in @@ -0,0 +1,10054 @@ +110 +2 0 +1 0 +1 1 +2 0 +1 1 +1 0 +2 1 +1 1 +1 0 +3 0 +1 1 +3 1 +2 0 +3 1 +1 1 +3 1 +2 0 +6 2 +6 1 +9 1 +5 0 +3 1 +4 0 +8 1 +20 4 +7 1 +3 0 +10 0 +13 1 +4 1 +2 1 +12 1 +4 1 +3 1 +14 1 +2 0 +8 0 +13 0 +8 1 +5 1 +13 1 +12 1 +13 0 +2 0 +6 1 +10 10 +1 0 +2 0 +3 0 +4 0 +5 0 +6 0 +7 0 +8 0 +9 0 +10 0 +20 2 +12 1 +16 1 +15 1 +21 1 +27 1 +16 0 +5 1 +23 0 +16 0 +19 1 +9 1 +16 1 +8 0 +16 1 +25 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+2 0 +42 0 +53 0 +30 1 +34 0 +16 1 +30 1 +43 0 +66 0 +54 0 +9 1 +22 1 +5 1 +59 0 +51 1 +10 1 +59 0 +22 0 +37 1 +61 0 +59 0 +34 1 +31 1 +37 1 +3 0 +44 0 +6 0 +40 0 +52 1 +26 0 +23 0 +39 0 +51 0 +63 0 +55 0 +6 1 +11 0 +58 0 +42 0 +44 0 +2 1 +55 0 +44 0 +19 1 +57 0 +23 1 +3 1 +53 0 +15 1 +29 0 diff --git a/2020/round3/railroad_renovations.md b/2020/round3/railroad_renovations.md new file mode 100644 index 0000000000000000000000000000000000000000..ff784fc3c3936c95e936e3aeddb5d0a890b57be7 --- /dev/null +++ b/2020/round3/railroad_renovations.md @@ -0,0 +1,43 @@ +Pennington has taken \(N\) trips on his favourite railroad line, which is 1 billion metres long. + +When the line was first built, creaky wood was used to support the tracks, which made a terrible racket as the train ran along them. Pennington is aware that, since then, 0 or more renovations have been made to the line at various points in time (either before his first trip or in between some of his trips), each of which would have involved replacing the supports along some continuous, possibly-degenerate interval of the line with integer endpoints (for example, the intervals \([1m, 2m]\) or \([3m, 3m]\) could have been renovated, but not \([3.5m, 4.5m]\)). No renovations were made *during* any of his trips. No two renovations' intervals could have inclusively overlapped with one another (for example, if one renovation involved the interval of positions \([5m, 10m]\), then there could not have been another one involving \([6m, 8m]\) nor \([10m, 11m]\), but there could have been one involving \([11m, 12m]\)). Renovations would never have been undone; each one had a permanent effect on its interval of the line. + +In general, Pennington just sleeps the entire time he's on the train (it is a long ride, after all). However, he did briefly wake up exactly once per trip he's taken. During his \(i\)th trip, he woke up while the train was at position \(P_i\) metres along the line, and he noticed whether the supports at that position had been renovated by that point in time — if \(R_i = 1\), they had been, while if \(R_i = 0\), they had not been. + +Pennington suspects he might have misremembered the order of his trips, though. To get their actual order, it may be necessary to take up to \(K\) of them and insert them into other positions in the ordered list of trips (with each one moved to either before the first trip, between any two trips, or after the final trip). For example, if \(K = 1\), he could rearrange the 2nd trip to have actually occurred between the 4th and 5th trips (essentially shifting trips 3 and 4 to have been his 2nd and 3rd trips). It's possible that his observations would only be consistent with any possible sequence of renovations after some rearrangements of this sort, or even that they would be inconsistent after any possible set of \(K\) rearrangements. + +What's the minimum possible number of renovations that could have been made by the end of Pennington's final trip which would have been consistent with all \(N\) of his observations (after rearranging at most \(K\) of the trips), if this would have been possible at all? + +# Constraints + +\(1 \le T \le 95\) +\(1 \le N \le 500\) +\(0 \le K \le N\) +\(1 \le P_i \le 1,000,000,000\) +\(0 \le R_i \le 1\) + +The sum of \(N\) across all sets of train trips is at most 10,000. + +# Input + +Input begins with an integer \(T\), the number of sets of train trips. For each set there are \(N + 1\) lines. The first line contains the 2 space-separated integers \(N\) and \(K\). \(N\) lines follow, the \(i\)th of which contains the 2 space-separated integers \(P_i\) and \(R_i\). + + +# Output + +For each set of train trips, print a line containing *"Case #i: "*, followed by a single integer, the minimum possible number of renovations which might have been made by the end of Pennington's final trip, or \(-1\) if no set of renovations would be consistent with his observations, after at most \(K\) trip rearrangements. + + + + +# Explanation of Sample + +In the first case, there could have been a single renovation involving the interval \([1m, 1m]\) between Pennington's first and second trips. + +In the second case, no sequence of renovations could explain the supports at position \(1m\) having been renovated by his first trip but no longer renovated by his second trip. + +In the third case, if his first trip is rearranged to occur after his second trip, then a single renovation is once again possible as in the first case. + +In the fourth case, there could have been two renovations involving the intervals \([1m, 1m]\) and \([3m, 3m]\) before any of his trips. It's impossible for fewer than two renovations to have occurred. + +In the fifth case, if his third trip is rearranged to occur before his first trip, then a single renovation involving the interval \([1m, 3m]\) could have occurred just after it. diff --git a/2020/round3/railroad_renovations.out b/2020/round3/railroad_renovations.out new file mode 100644 index 0000000000000000000000000000000000000000..e3495f7ab36dbec34cb4d20f3704c6d3b3a40680 --- /dev/null +++ b/2020/round3/railroad_renovations.out @@ -0,0 +1,110 @@ +Case #1: 1 +Case #2: -1 +Case #3: 1 +Case #4: 2 +Case #5: 1 +Case #6: 1 +Case #7: 4 +Case #8: 0 +Case #9: 7 +Case #10: -1 +Case #11: 4 +Case #12: 1 +Case #13: 4 +Case #14: 2 +Case #15: 1 +Case #16: 1 +Case #17: 1 +Case #18: 250 +Case #19: 29 +Case #20: 12 +Case #21: -1 +Case #22: 9 +Case #23: 5 +Case #24: 2 +Case #25: -1 +Case #26: 6 +Case #27: 2 +Case #28: 1 +Case #29: 1 +Case #30: 1 +Case #31: 7 +Case #32: 1 +Case #33: 6 +Case #34: -1 +Case #35: -1 +Case #36: 1 +Case #37: -1 +Case #38: 1 +Case #39: 1 +Case #40: 1 +Case #41: 9 +Case #42: 3 +Case #43: 1 +Case #44: 2 +Case #45: 1 +Case #46: 4 +Case #47: 10 +Case #48: 3 +Case #49: 10 +Case #50: 9 +Case #51: 8 +Case #52: 11 +Case #53: 5 +Case #54: 17 +Case #55: 9 +Case #56: 8 +Case #57: 13 +Case #58: 12 +Case #59: 7 +Case #60: 10 +Case #61: 10 +Case #62: 15 +Case #63: 8 +Case #64: 5 +Case #65: 3 +Case #66: 2 +Case #67: 6 +Case #68: 7 +Case #69: 1 +Case #70: 4 +Case #71: 2 +Case #72: 5 +Case #73: 10 +Case #74: 4 +Case #75: 9 +Case #76: 1 +Case #77: 4 +Case #78: 1 +Case #79: 5 +Case #80: 12 +Case #81: 2 +Case #82: 6 +Case #83: 2 +Case #84: 5 +Case #85: 3 +Case #86: 8 +Case #87: 8 +Case #88: 6 +Case #89: 3 +Case #90: 5 +Case #91: 2 +Case #92: 4 +Case #93: 3 +Case #94: 4 +Case #95: 4 +Case #96: 6 +Case #97: 11 +Case #98: 5 +Case #99: 9 +Case #100: 6 +Case #101: 3 +Case #102: 9 +Case #103: 3 +Case #104: 1 +Case #105: 5 +Case #106: 3 +Case #107: 3 +Case #108: 10 +Case #109: 5 +Case #110: 2 diff --git a/2020/round3/railroad_renovations_sol.md b/2020/round3/railroad_renovations_sol.md new file mode 100644 index 0000000000000000000000000000000000000000..f24b21b02d37dbcd44aa9cfe11790483afa6dff5 --- /dev/null +++ b/2020/round3/railroad_renovations_sol.md @@ -0,0 +1,21 @@ +We'll begin by sorting the observations in non-decreasing order of position (ordering equal-position observations arbitrarily), and re-numbering them in this order. In addition to each observation \(i\)'s \(P_i\) and \(R_i\) values, we'll store the original trip index \(I_i\) which it corresponds to. + +We'll proceed with dynamic programming. Let \(D_{i,j}\) be the minimum number of renovations which could have occurred if we consider only the first \(i\) observations in this ordered list, such that \(j\) of their trips have been rearranged. We'll only concern ourselves with indices \(i\) on boundaries between different positions (specifically, such that \(i = 0\), \(i = N\), or \(P_i < P_{i+1}\)). We know that \(D_{0,0} = 0\), and our answer will be \(\min(D_{N,0..K})\). + +In order to compute \(D_{i,j}\), we'll consider each possible earlier index \(k\) such that \(k < i\) and either \(k = 0\) or \(P_k < P_{k+1}\), determine what can be done with observations \((k+1)..i\), compute a candidate value for \(D_{i,j}\) accordingly, and take the minimum of these candidate values. + +If \(R_o = 0\) for each \(o \in [k+1, i]\), then no renovations nor trip rearrangements are required, and our candidate value will simply be \(D_{k,j}\). + +Otherwise, there must be at least one renovation amongst these observations' positions, and we'll assume that there was exactly one renovation spanning that full interval of positions (noting that other possibilities would be captured by different choices of \(k\)). We must then determine the minimum number of trip rearrangements necessary for that to be possible. + +Considering each possible original trip index \(t\) just after which the renovation was made (such that \(0 \le t \le N\), with \(t = 0\) representing the renovation having been before the first original trip), it's optimal to: + +- Rearrange all 1-observations before that time (observations \(o\) such that \(k+1 \le o \le i\), \(R_o = 1\), and \(I_i \le t\)) to have occurred after that time +- Rearrange all 0-observations after that time (observations \(o\) such that \(k+1 \le o \le i\), \(R_o = 0\), and \(I_i > t\)) to have occurred before that time + +Given the minimum number \(x\) of observations which need to be rearranged (across all possible choices of \(t\)), our candidate value for \(D_{i,j}\) will then be \(D_{k,j-x} + 1\) (or \(\infty\) if \(j-x < 0\)). + +The most direct, naive implementation of the above approach would be too slow, taking \(O(N^5)\) time (considering \(O(N^4)\) quadruples of \((i, j, k, t)\), and iterating over \(O(N)\) observations each time to count the number of trips needing rearrangement). However, the time complexity may be reduced to \(O(N^3)\) through a combination of two optimizations each removing a factor of \(N\): + +1. We can observe that the value \(x\) described above is independent of \(j\), meaning that we only need to compute it once per pair of \((i, k)\). +2. For each pair of \((i, k)\), if we sweep over all observations in non-decreasing order of \(I\) value (considering only ones in that interval of positions), while keeping track of the number of 0-observations and 1-observations already encountered. This is sufficient to compute \(x\) with no additional time. diff --git a/2020/round3/smart_carts.cpp b/2020/round3/smart_carts.cpp new file mode 100644 index 0000000000000000000000000000000000000000..857cf9fd4b3a2875f6bda2b1f236af0b8a1b86fb --- /dev/null +++ b/2020/round3/smart_carts.cpp @@ -0,0 +1,264 @@ +// Smart Carts +// Solution by Jacob Plachta + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using namespace std; + +#define LL long long +#define LD long double +#define PR pair + +#define Fox(i,n) for (i=0; i=0; i--) +#define FoxR1(i,n) for (i=n; i>0; i--) +#define FoxRI(i,a,b) for (i=b; i>=a; i--) +#define Foxen(i,s) for (i=s.begin(); i!=s.end(); i++) +#define Min(a,b) a=min(a,b) +#define Max(a,b) a=max(a,b) +#define Sz(s) int((s).size()) +#define All(s) (s).begin(),(s).end() +#define Fill(s,v) memset(s,v,sizeof(s)) +#define pb push_back +#define mp make_pair +#define x first +#define y second + +template T Abs(T x) { return(x < 0 ? -x : x); } +template T Sqr(T x) { return(x * x); } +string plural(string s) { return(Sz(s) && s[Sz(s) - 1] == 'x' ? s + "en" : s + "s"); } + +const int INF = (int)1e9; +const LD EPS = 1e-12; +const LD PI = acos(-1.0); + +#define GETCHAR getchar_unlocked + +bool Read(int& x) +{ + char c, r = 0, n = 0; + x = 0; + for (;;) + { + c = GETCHAR(); + if ((c < 0) && (!r)) + return(0); + if ((c == '-') && (!r)) + n = 1; + else + if ((c >= '0') && (c <= '9')) + x = x * 10 + c - '0', r = 1; + else + if (r) + break; + } + if (n) + x = -x; + return(1); +} + +#define LIM 705 + +int N; +int nxt[2][LIM], prv[2][LIM]; +vector seq[2][2]; +int L[2][2]; +PR pos[2][LIM]; + +int F[2], subS[2]; +vector CS; +int csInd, sumC, baseAns; + +int ProcessQuery0(int* K) +{ + int i, ans = 0; + // check whether either initial line is too long to be valid + Fox(i, 2) + if (L[0][i] > K[i]) + return(-1); + // count initially-satisfied carts + Fox(i, N) + ans += nxt[0][i] == nxt[1][i]; + return(ans); +} + +void PrecomputeForQuery1(int C) +{ + int i, j; + // determine which carts are free + Fox(i, 2) + F[i] = min(C - L[0][1 - i], L[0][i]); + auto IsFree = [&](int i, int e) { // if e=1, include non-free, accessible latches + PR p = pos[0][i]; + return(p.y > L[0][p.x] - F[p.x] - e); + }; + // compute number of potentially-satisfiable carts + baseAns = 0; + Fox(i, N) + baseAns += nxt[0][i] == nxt[1][i] || // initially satisfied? + (IsFree(i, 0) && IsFree(nxt[1][i], 1)); // free, and target accessible? + // split free, potentially-satisfiable carts into chains + // and compute achievable subset sums of chain lengths + sumC = 0; + Fill(subS, 0); + bitset BC = 1; + Fox(i, 2) + { + auto& s = seq[1][i]; + int c = 0, f = -1; + Fox(j, Sz(s)) + { + if (!IsFree(s[j], 0)) + continue; + c++; + // chain forced to belong to a certain line? + if (c == 1 && j && IsFree(s[j - 1], 1)) + f = pos[0][s[j - 1]].x; + // chain ends with this cart? + if (j + 1 == Sz(s) || !IsFree(s[j + 1], 0)) + { + if (f < 0) + sumC += c, BC |= (BC << c); + else + subS[f] += c; + c = 0, f = -1; + } + } + } + // store results + CS.clear(); + Fox(i, sumC + 1) + if (BC[i]) + CS.pb(i); + csInd = 0; +} + +int ProcessQuery1(int* K) +{ + // compute space for free carts in each line + int i, S[2]; + Fox(i, 2) + S[i] = K[i] - (L[0][i] - F[i]) - subS[i]; + // check whether all free cart chains may be packed into the lines + if (min(S[0], S[1]) < 0) + return(baseAns - 1); + while (csInd + 1 < Sz(CS) && CS[csInd + 1] <= S[0]) + csInd++; + if (sumC - CS[csInd] <= S[1]) + return(baseAns); + return(baseAns - 1); // 1 cart must remain unsatisfied due to chain breakage +} + +int ProcessQuery2(int* K) +{ + int i; + // check whether either target line is too long to be fully satisfied + Fox(i, 2) + if (L[1][i] > K[i]) + return(N - 1); + return(N); +} + +int ProcessQuery(int d, int c, int x, int y) +{ + int i, K[2] = { min(c, x), min(c, y) }; + // check whether either initial line is too long to be valid + Fox(i, 2) + if (L[0][i] > c) + return(-1); + // check whether there are too many carts to form valid final lines + if (N > K[0] + K[1]) + return(-1); + return d == 0 + ? ProcessQuery0(K) + : d == 1 + ? ProcessQuery1(K) + : ProcessQuery2(K); +} + +void ProcessCase() +{ + int i, j, z; + // init + Fill(prv, -1); + // input + Read(N); + Fox(i, N) + { + Fox(z, 2) + { + Read(nxt[z][i]), nxt[z][i]--; + prv[z][nxt[z][i]] = i; + } + } + // split into lines (initial/target) + Fox(z, 2) + { + Fox(i, 2) + { + seq[z][i].clear(); + j = N + i; + while (j >= 0) + { + pos[z][j] = mp(i, Sz(seq[z][i])); + seq[z][i].pb(j); + j = prv[z][j]; + } + L[z][i] = Sz(seq[z][i]) - 1; + } + } + // process queries + int d, c, x, y; + LL G[LIM] = { 0 }; + Fox(d, min(3, N + 1)) + { + Fox(c, N + 1) + { + if (d == 1) + PrecomputeForQuery1(c); + Fox(x, N + 1) + { + Fox(y, N + 1) + G[ProcessQuery(d, c, x, y) + 1] += d == 2 ? N - 1 : 1; + } + } + } + Fox(i, N + 2) + printf(" %lld", G[i]); + printf("\n"); +} + +int main() +{ + int T, t; + Read(T); + Fox1(t, T) + { + printf("Case #%d:", t); + ProcessCase(); + } + return(0); +} \ No newline at end of file diff --git a/2020/round3/smart_carts.in b/2020/round3/smart_carts.in new file mode 100644 index 0000000000000000000000000000000000000000..f510ee5cad9bc9bda531ecb8b9c3297054db91ce --- /dev/null +++ b/2020/round3/smart_carts.in @@ -0,0 +1,7081 @@ +108 +1 +2 3 +3 +4 5 +1 4 +5 1 +4 +3 2 +4 6 +5 5 +1 3 +6 +5 6 +3 5 +4 7 +6 2 +7 1 +8 3 +10 +11 2 +1 3 +2 4 +3 5 +4 6 +12 7 +6 8 +7 9 +8 10 +9 11 +10 +12 12 +1 1 +2 2 +3 3 +4 4 +5 5 +6 6 +7 7 +8 8 +9 9 +51 +28 33 +12 49 +20 38 +11 12 +27 44 +25 4 +6 52 +30 31 +21 45 +32 28 +22 20 +14 2 +4 42 +10 41 +29 30 +45 18 +15 32 +23 50 +17 47 +47 25 +13 43 +40 3 +16 48 +8 26 +38 40 +48 51 +2 37 +51 39 +26 7 +41 14 +36 29 +43 1 +3 11 +42 36 +49 6 +52 53 +19 13 +50 21 +33 23 +34 34 +9 46 +53 8 +44 24 +39 10 +35 5 +18 19 +7 27 +46 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+19 12 +14 14 +3 10 +6 3 +64 +54 20 +43 25 +66 47 +14 36 +42 66 +21 13 +52 12 +31 58 +53 51 +4 26 +29 42 +48 1 +39 63 +15 57 +62 17 +1 55 +10 18 +6 48 +47 21 +64 23 +55 6 +32 56 +24 52 +51 38 +2 39 +35 14 +46 54 +17 37 +38 32 +58 44 +3 65 +12 7 +13 27 +56 33 +20 50 +40 30 +11 3 +61 4 +5 53 +7 15 +45 34 +57 28 +49 9 +26 60 +36 61 +28 40 +63 16 +60 41 +30 8 +65 19 +18 24 +22 46 +27 43 +34 2 +8 49 +33 31 +23 5 +19 29 +37 10 +9 35 +16 22 +59 59 +41 45 +50 62 diff --git a/2020/round3/smart_carts.md b/2020/round3/smart_carts.md new file mode 100644 index 0000000000000000000000000000000000000000..4e323c503e52f42e6157b4a48a3017171bb55caf --- /dev/null +++ b/2020/round3/smart_carts.md @@ -0,0 +1,92 @@ +Every week you take a socially-distanced trip to Overwaitea, your favourite grocery store, to stock up on food. Out front, there are \(N\) shopping carts, numbered from \(1\) to \(N\). To dissuade theft, the carts are locked together and you need to insert a quarter to unlock one cart from the rest. When you return your cart, you get your quarter back. Now that you think about it, maybe 25 Canadian cents is actually a pretty good deal for your very own shopping cart... + +Each cart has a connecting chain leading out of it. There are also \(N+2\) latches, numbered from \(1\) to \(N+2\), which chains may be connected to (with each latch having at most one incoming chain). Each of the first \(N\) latches can be found on carts, with latch \(i\) fixed to cart \(i\). The remaining two latches (latches \(N+1\) and \(N+2\)) are instead mounted to a wall. Each cart \(i\)'s chain is initially connected to latch \(A_i\), but you suspect it would be more aesthetically pleasing if it was instead attached to an ideal target latch \(B_i\). + +A *sequence* of \(K\) carts \(S_{1..K}\) is one in which latch \(S_1\) has no incoming chain, cart \(S_1\)'s chain connects to latch \(S_2\), cart \(S_2\)'s chain connects to latch \(S_3\), and so on, with cart \(S_{K-1}\)'s chain connecting to latch \(S_K\), and cart \(S_K\)'s chain not connecting to any cart (but rather to either a wall latch or no latch). + +The carts are rigid enough that it's impossible to ever form a "cycle" of carts, such that repeatedly following chain connections forwards from an initial cart will result in returning to that initial cart. This means that each cart is always part of exactly one *sequence*, and is present in that *sequence* exactly once. + +The values of \(A_{1..N}\) and \(B_{1..N}\) are given such that the carts initially form a valid configuration (with no "cycles"), and would also form a valid configuration if every cart's chain were to be connected to its ideal target latch. + +For example, if \(N = 3\) and \(A = [4, 1, 5]\), the initial configuration looks as follows (having two *sequences* of carts \([2, 1]\) and \([3]\)): + +{{PHOTO_ID:1217734278611436}} + +You may repeatedly perform either of the following actions before leaving the carts in a desired *final state*, in which every chain must end up once again connected to a latch: + + 1. Pay 1 quarter (if you have at least 1) to detach any given cart's chain from whichever latch it's currently connected to (if any) + 2. Gain 1 quarter by attaching any given cart's currently-unattached chain to a latch (as long as no other chain is currently connected to that latch, and a "cycle" of carts would not be formed) + +For example, beginning with the above configuration and with at least 1 quarter in hand, you could perform **action 1** to detach cart 1's chain and produce the following configuration (which is *not* a valid *final state* due to cart 1's chain not being connected to any latch): + +{{PHOTO_ID:3302289739857297}} + +You could then perform **action 2** to attach cart 1's chain to cart 3's latch (forming a valid *final state*): + +{{PHOTO_ID:766911834140516}} + +Consider a hypothetical scenario in which: + + - You begin with \(Q\) quarters + - A *sequence* consisting of more than \(C\) carts must never be present (including initially) + - The *final state* must not have a *sequence* consisting of more than \(X\) carts attached to latch \(N+1\) + - The *final state* must not have a *sequence* consisting of more than \(Y\) carts attached to latch \(N+2\) + +Let \(F(Q, C, X, Y)\) be the maximum number of chains which can end up connected to their ideal target latches in a *final state* given such a scenario, or \(-1\) if it's impossible to achieve a valid *final state* at all. + +Let \(G(v)\) be the number of quadruples \((Q, C, X, Y)\) such that \(0 \le Q, C, X, Y \le N\) and \(F(Q, C, X, Y) = v\). You'd like to compute \(G(v)\) for each integer \(v\) such that \(-1 \le v \le N\). + +# Constraints + +\(1 \le T \le 85\) +\(1 \le N \le 700\) +\(1 \le A_i, B_i \le N+2\) + +The sum of \(N\) across all test cases is at most 7000. + +# Input + +Input begins with an integer \(T\), the number of sets of shopping carts. For each set there are \(N+1\) lines. The first contains the integer \(N\). \(N\) lines follow, the \(i\)th of which contains the space-separated integers \(A_i\) and \(B_i\). + + + + +# Output + +For each set of shopping carts, print a line containing *"Case #i: "*, followed by \(N+2\) integers, \(G(-1)\), \(G(0)\), \(G(1)\), ..., \(G(N-1)\), \(G(N)\), as defined above. + + +# Explanation of Sample + +In the first case, the initial configuration is as follows: + +{{PHOTO_ID:3371452566235882}} + +There are \(2^4 = 16\) quadruples \((Q, C, X, Y)\) to consider. For 11 of them, \(F(Q, C, X, Y) = -1\), while the following 5 yield different results: + +- \(F(0, 1, 1, 0) = 0\) +- \(F(0, 1, 1, 1) = 0\) +- \(F(1, 1, 1, 0) = 0\) +- \(F(1, 1, 0, 1) = 1\) +- \(F(1, 1, 1, 1) = 1\) + +Several evaluations of \(F(Q, C, X, Y)\) are explained below: + +- \(F(1, 0, 1, 1) = -1\) because the initial configuration features a *sequence* of 1 cart, which is disallowed due to \(C = 0\). +- \(F(1, 1, 0, 0) = -1\) because every possible *final state* features a *sequence* of 1 cart connected to either latch 2 or latch 3, which is disallowed due to \(X = Y = 0\). +- \(F(0, 1, 1, 1) = 0\) because no action may initially taken without at least 1 quarter. +- \(F(1, 1, 0, 1) = 1\) because you may spend your 1 quarter to disconnect cart 1's chain from latch 2, and then gain your quarter back while connecting it to latch 3 (its ideal target latch). + +In the second case, the initial configuration is as follows: + +{{PHOTO_ID:1217734278611436}} + +Several evaluations of \(F(Q, C, X, Y)\) are explained below: + +- \(F(3, 2, 2, 2) = 3\) because you may spend your 3 quarters disconnecting all of the carts' chains, and then gain your quarters back while connecting the carts' chains to their ideal target latches in any order (without a *sequence* of more than 2 carts ever being present). This yields the following *final state*: + +{{PHOTO_ID:2978496958918891}} + +- \(F(3, 3, 3, 1) = 2\) because you may similarly disconnect and then reconnect all of the carts' chains, but cannot arrive at the same *final state* as above (as it features a *sequence* of 2 carts connected to latch 5, which is disallowed due to \(Y = 1\)). This means that only up to 2 chains may end up connected to their ideal target latches instead, for example as in the following *final state*: + +{{PHOTO_ID:3274155842706395}} diff --git a/2020/round3/smart_carts.out b/2020/round3/smart_carts.out new file mode 100644 index 0000000000000000000000000000000000000000..4142caec6e8af16a8d05b2d4775b586eba2d438a --- /dev/null +++ b/2020/round3/smart_carts.out @@ -0,0 +1,108 @@ +Case #1: 11 3 2 +Case #2: 190 20 0 16 30 +Case #3: 560 0 5 0 24 36 +Case #4: 1900 67 0 7 19 0 366 42 +Case #5: 11165 298 2 52 6 54 0 20 44 0 2890 110 +Case #6: 13970 0 0 0 0 0 0 0 0 0 550 121 +Case #7: 6028516 0 13200 0 0 20 978 0 60 976 1072 52 1070 1122 1168 14 2404 0 37 1209 0 36 1232 92 1196 19 1287 0 0 158 1164 7 1329 0 0 106 1242 0 0 218 1140 0 0 470 896 618 754 0 762 614 0 991816 254562 +Case #8: 172377204066 28487260 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 160967 160443 783 322965 325296 513 327089 254 164310 253 164817 757 165068 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/dev/null +++ b/2020/round3/smart_carts_sol.md @@ -0,0 +1,29 @@ +As a general observation, we can note that the carts in both the initial configuration and any valid *final state* form at most two *sequences*, one attached to latch \(N+1\) and the other to latch \(N+2\). We'll refer to these as the first and second "lines" (either of which may be empty), with the start of each line being the cart attached to the wall latch. We'll then let \(L_i\) be the length of the \(i\)th initial line, and \(K_i\) be the cap on the length of the \(i\)th line in the *final state* (\(K = [X, Y]\)). + +Now, we'll begin by discussing how to evaluate \(F(Q, C, X, Y)\), before extending that to efficiently aggregate values across all quadruples \((Q, C, X, Y)\). + +Checking for impossibility (\(F(Q, C, X, Y) = -1\)) is relatively simple. It's impossible if \(C\) is smaller than either \(L_1\) or \(L_2\), and impossible if \(\min(C, X) + \min(C, Y) < N\). If \(Q=0\), then it's also impossible if \(K_i < L_i\) for either line \(i\). It's otherwise possible. + +Assuming it is possible, there are three distinct cases to consider based on the value of \(Q\): + +1. When \(Q=0\), the answer is simply the number of initially-satisfied carts (carts \(i\) such that \(A_i = B_i\)). +2. When \(Q>1\), we can show that it's possible to arrange the carts into any *final state* which satisfies the basic requirements (that is, such that length of each line \(i\) does not exceed \(\min(C, K_i)\)). +One way of demonstrating this is as follows. It's possible to move any cart \(i\) to the end of its current line without involving a longer *sequence* (by disconnecting cart \(i\) from the previous/next latches, connecting it to the former ending cart, and then connecting the line back together). Through repeated applications of this, it's possible to arbitrarily permute the carts in a line. It's furthermore trivially possible to move one line's ending cart to the end of the other line, or to swap the ending carts of both lines, similarly without involving longer *sequences* than necessary. A combination of these operations allows all \(N\) carts to be arbitrarily rearranged. +As a result of this fact, if we consider the lines formed by all chains being connected to their ideal target latches, the answer is simply \(N\) if those lines' lengths satisfy these basic requirements, or \(N-1\) otherwise (as we'll need to split one line at one point, attaching its suffix to the other line instead, in order to make everything fit). +3. When \(Q=1\), things are more involved; we'll discuss this case below. + +Let \(F_i = \min(C - L_{3-i}, L_i)\). The last \(F_i\) carts in the \(i\)th line are free to be moved (either amongst themselves or to the other line), while the remaining \(L_i - F_i\) carts are stuck and can never be moved. All rearrangements of the \(F_1 + F_2\) free carts across both lines are achievable (as long as the length of each line \(i\) in the *final state* does not exceed \(\min(C, K_i)\)), which can be demonstrated similarly to in the \(Q>1\) case above. The \((F_i+1)\)th latch from the end of the line, though stuck, is accessible to have a new chain attached to it. + +Each cart \(i\) is therefore "potentially satisfiable" if at least one of the following conditions is met: + +- It's initially satisfied (\(A_i = B_i\)) +- It's free (is amongst the last \(F_x\) carts in its line \(x\)) and latch \(B_i\) is amongst the last \(F_y+1\) latches in its line \(y\) (regardless of whether or not \(x = y\)) + +However, similarly to the \(Q>1\) case, not all potentially satisfiable carts might actually be simultaneously satisfiable due to restrictions on the *final state*'s line lengths. In order to determine whether this is the case, we'll represent the free, potentially satisfiable carts as a set of "blocks", with each block consisting of 1 or more such carts which should be contiguously connected together in the *final state*. If the ideal target latch of a block's frontmost cart is stuck, then that block is forced to be placed in that latch's particular line. Any other blocks may be rearranged arbitrarily between the two lines, as long as the carts in each block remain together in a single line. Therefore, all potentially satisfiable carts may be satisfied if and only if it's possible to pack these blocks into the two lines without exceeding their maximum allowed lengths, which can be determined if we compute all achievable subset sums of block lengths. If this can't be done, then the answer should be reduced by \(1\) instead, as one block will need to be split up at one point in order to make everything fit. + +We're now ready to discuss how to aggregate \(F(Q, C, X, Y)\) values across all quadruples \((Q, C, X, Y)\). We'll shoot for a time complexity of \(O(N^3)\), which is sufficient, though slightly better time complexities (e.g. \(O(N^2 \sqrt{N})\)) are also achievable through some optimizations not discussed in detail here. + +Assuming we precompute basic information around the initial and target lines (e.g. \(L_{1..2}\)), it then takes only constant time to evalate \(F(Q, C, X, Y)\) when \(Q \ne 1\). Combined with the fact that \(F(Q, C, X, Y) = F(2, C, X, Y)\) when \(Q > 1\), we can handle all quadruples \((Q, C, X, Y)\) such that \(Q \ne 1\) by simply iterating over all \(O(N^3)\) triples \((C, X, Y)\) (though it's also possible to use the fact that many are evaluated similarly). + +For the \(Q = 1\) case, we can observe that the computation of potentially satisfiable carts and achievable block length subset sums depends only on \(C\) (not on \(X\) and \(Y\)). We therefore only need to go through that process \(O(N)\) times, meaning that a naive \(O(N^2)\) subset sum implementation is sufficient (though it's possible in as little as \(O(N \sqrt{N})\) time). Then, given an ordered list of achievable block length subset sums for each value of \(C\), we can iterate over all pairs \((X, Y)\) in another \(O(N^2)\) time and check which ones will have their answer reduced by \(1\) with no additional time (though it's also possible to only iterate over values of \(X\) and deal with the \(Y\) values using binary search). +