2020/finals/cake_cutting_committee.cpp

// Cake-Cutting Committee
// Solution by Jacob Plachta

#include <algorithm>
#include <functional>
#include <numeric>
#include <iostream>
#include <iomanip>
#include <cstdio>
#include <cmath>
#include <complex>
#include <cstdlib>
#include <ctime>
#include <cstring>
#include <cassert>
#include <string>
#include <vector>
#include <list>
#include <map>
#include <set>
#include <unordered_map>
#include <unordered_set>
#include <deque>
#include <queue>
#include <stack>
#include <bitset>
#include <sstream>
using namespace std;

#define LL long long
#define LD long double
#define PR pair<int,int>

#define Fox(i,n) for (i=0; i<n; i++)
#define Fox1(i,n) for (i=1; i<=n; i++)
#define FoxI(i,a,b) for (i=a; i<=b; i++)
#define FoxR(i,n) for (i=(n)-1; 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<typename T> T Abs(T x) { return(x < 0 ? -x : x); }
template<typename T> 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<int> h)

{
  int i, j, s;
  // compare all pieces against dividing line, and assemble line sweep events
  int base = 0;
  vector<int> CY;
  vector<Event> 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);
}