#include #include #include #include #include #include using namespace std; using LL = long long; struct Point { LL x, y; Point() : x(0), y(0) {} Point(LL _x, LL _y) : x(_x), y(_y) {} bool operator==(const Point &p) const { return x == p.x && y == p.y; } bool operator!=(const Point &p) const { return x != p.x || y != p.y; } bool operator<(const Point &p) const { return x != p.x ? x < p.x : y < p.y; } }; struct Line { Point left, right; Line() : left() {} Line(const Point &p1, const Point &p2) : left(p1), right(p2) { if (left.x > right.x) { swap(left, right); } } bool operator==(const Line &l) const { return left == l.left && right == l.right; } bool is_horizontal() const { return left.y == right.y; } bool is_vertical() const { return left.x == right.x; } }; mt19937 rng(chrono::steady_clock::now().time_since_epoch().count()); struct Treap { Line *key; Treap *left, *right; int weight; Treap() : key(nullptr), left(nullptr), right(nullptr), weight(rng()) {} Treap(Line *l) : key(l), left(nullptr), right(nullptr), weight(rng()) {} }; static Treap *rotate_left(Treap *x) { // assert(x && x->left); Treap *y = x->left; x->left = y->right; y->right = x; return y; } static Treap *rotate_right(Treap *x) { // assert(x && x->right); Treap *y = x->right; x->right = y->left; y->left = x; return y; } template static Treap *persistent_insert(Treap *h, Line *key, const TCompare &cmp) { if (!h) { return new Treap(key); } Treap *x = new Treap(*h); if (cmp(key, x->key)) { x->left = persistent_insert(x->left, key, cmp); if (x->left->weight < x->weight) { return rotate_left(x); } } else if (cmp(x->key, key)) { x->right = persistent_insert(x->right, key, cmp); if (x->right->weight < x->weight) { return rotate_right(x); } } else { // assert(false); // Already inserted. } // assert(x != nullptr); return x; } static Treap *persistent_remove(Treap *h) { if (h->left != nullptr && h->right != nullptr) { Treap *y; if (h->left->weight < h->right->weight) { h->left = new Treap(*h->left); y = rotate_left(h); y->right = persistent_remove(h); } else { h->right = new Treap(*h->right); y = rotate_right(h); y->left = persistent_remove(h); } return y; } Treap *next = h->left; if (h->right != nullptr) { next = h->right; } // delete h; return next; } template static Treap *persistent_remove(Treap *h, Line *key, const TCompare &cmp) { // assert(h != nullptr); Treap *x = new Treap(*h); if (cmp(key, x->key)) { x->left = persistent_remove(x->left, key, cmp); } else if (cmp(x->key, key)) { x->right = persistent_remove(x->right, key, cmp); } else { return persistent_remove(x); } return x; } using TreapPair = pair; template TreapPair floor_and_ceil(Treap *h, const Point &p, const TCompare &cmp) { Treap *x = h; TreapPair result(nullptr, nullptr); while (x) { if (cmp(p, x->key)) { result.second = x; x = x->left; } else { result.first = x; x = x->right; } } return result; } template pair, vector> get_lower_and_upper_hull(It lo, It hi) { auto cross = [&](const Point &a, const Point &b, const Point &o) { return (a.x - o.x)*(b.y - o.y) - (a.y - o.y)*(b.x - o.x); }; int len = distance(lo, hi), k = 0; // assert(len >= 3); vector hull(2 * len); sort(lo, hi); for (It it = lo; it != hi; ++it) { while (k >= 2 && cross(hull[k - 1], *it, hull[k - 2]) >= 0) { k--; } hull[k++] = *it; } int t = k; for (It it = hi - 2; it != lo - 1; --it) { while (k > t && cross(hull[k - 1], *it, hull[k - 2]) >= 0) { k--; } hull[k++] = *it; } hull.resize(k); vector lower, upper; for (int i = 1; i < t; i++) { upper.push_back(new Line(hull[i - 1], hull[i])); } for (int i = t; i < k; i++) { lower.push_back(new Line(hull[i - 1], hull[i])); } return make_pair(lower, upper); } enum class EventType : int { Open = 1, Close = 2, }; struct Event { Line *line; EventType type; Event() : line(nullptr) {} Event(Line *_line, const EventType _type) : line(_line), type(_type) {} }; struct EnclosedPolygons { vector edges; unordered_map poly_id; unordered_map is_upper; unordered_map parent; vector xvals; vector events; Treap *tlatest; vector tversions; static double eval_line(const Line *l, LL x) { // assert(!l->is_vertical()); double m = (double)(l->right.y - l->left.y) / (l->right.x - l->left.x); double b = l->left.y - m * l->left.x; return m * x + b; } static bool cmp_point(const Point &p, const Line *l) { if (p == l->left || p == l->right) { return false; } return p.y < eval_line(l, p.x); } EnclosedPolygons(vector> &_polygons) : tlatest(nullptr) { for (size_t i = 0; i < _polygons.size(); ++i) { auto &points = _polygons[i]; for (const Point &p : points) { xvals.push_back(p.x); } auto hulls = get_lower_and_upper_hull(points.begin(), points.end()); auto add_edge = [&](Line *l) { edges.push_back(l); events.push_back(Event(l, EventType::Open)); events.push_back(Event(l, EventType::Close)); poly_id[l] = i; }; for (Line *l : hulls.first) { add_edge(l); } for (Line *l : hulls.second) { add_edge(l); is_upper[l] = true; } } sort(xvals.begin(), xvals.end()); xvals.resize(distance(xvals.begin(), unique(xvals.begin(), xvals.end()))); auto cmp_events = [&](const Event &a, const Event &b) { LL ax = a.type == EventType::Open ? a.line->left.x : a.line->right.x; LL ay = a.type == EventType::Open ? a.line->left.y : a.line->right.y; LL bx = b.type == EventType::Open ? b.line->left.x : b.line->right.x; LL by = b.type == EventType::Open ? b.line->left.y : b.line->right.y; if (ax != bx) { return ax < bx; } if (ay != by) { return ay < by; } // We sort Close events first, so we delete from the treap first, thus // avoiding comparing equal points. if (a.type != b.type) { return a.type == EventType::Close; } // When 2 open events to have the same point, it must be the vertex that // joins the upper and lower hull of the same polygon. We will sort the // one on the lower hull before. // assert(poly_id[a.line] == poly_id[b.line]); return ay < max(b.line->left.y, b.line->right.y); }; stable_sort(events.begin(), events.end(), cmp_events); parent[0] = -1; auto xit = xvals.begin(); auto eit = events.begin(); while (xit != xvals.end() && eit != events.end()) { LL curr_x = *xit; tversions.push_back(tlatest); // Returns whether l1 is below l2, assuming they don't intersect. auto cmp = [curr_x](const Line *l1, const Line *l2) { // First check if equal endpoints as we need a strict BST ordering. if (l1->left == l2->left) { int x = min(l1->right.x, l2->right.x); return eval_line(l1, x) < eval_line(l2, x); // edges meet at < } if (l1->right == l2->right) { int x = max(l1->left.x, l2->left.x); return eval_line(l1, x) < eval_line(l2, x); // edges meet at > } if (l1->right == l2->left) { // edges meet like --l1--x--l2-- return l1->left.y < l2->right.y; } if (l1->left == l2->right) { // else l1->left == l2->right and edges meet like --l2--x--l1-- return l1->right.y < l2->left.y; } return eval_line(l1, curr_x) < eval_line(l2, curr_x); }; // Process all events at this x, starting from the bottom. while (eit != events.end()) { if (eit->line->is_vertical()) { // Vertical edges are of no use. ++eit; continue; } if (curr_x != (eit->type == EventType::Open ? eit->line->left.x : eit->line->right.x)) { break; } int id = poly_id[eit->line]; if (parent.find(id) == parent.end()) { // If no parent is found yet... LL curr_y = eit->type == EventType::Open ? eit->line->left.y : eit->line->right.y; auto res = floor_and_ceil(tlatest, Point(curr_x, curr_y), cmp_point); Line *line_below = res.first ? res.first->key : nullptr; // assert(line_below != nullptr); int below_id = poly_id[line_below]; if (is_upper[line_below]) { parent[id] = parent[below_id]; } else { // If the line below is a lower hull edge, // then it must either be ourself or our parent. if (id != below_id) { parent[id] = below_id; } } } if (eit->type == EventType::Open) { tlatest = persistent_insert(tlatest, eit->line, cmp); } else { tlatest = persistent_remove(tlatest, eit->line, cmp); } ++eit; } ++xit; } } int get_containing_polygon(const Point &p) { // Search for an x. No need to consider 0, which is an empty treap. size_t left = 1, right = xvals.size() - 1; while (left + 1 < right) { int mid = (left + right) / 2; if (xvals[mid] <= p.x) { left = mid; } else { right = mid; } } if (left != right && xvals[left] <= p.x) { left = right; } // Query the line immediately below. auto res = floor_and_ceil(tversions[left], p, cmp_point); Line *line_below = res.first ? res.first->key : nullptr; // assert(line_below != nullptr); int below_id = poly_id[line_below]; return is_upper[line_below] ? parent[below_id] : below_id; } ~EnclosedPolygons() { for (Line *l : edges) { delete l; } } }; const size_t SEED = chrono::steady_clock::now().time_since_epoch().count(); size_t get_hash(size_t v) { size_t x = v + SEED; // http://xorshift.di.unimi.it/splitmix64.c x += 0x9e3779b97f4a7c15; x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9; x = (x ^ (x >> 27)) * 0x94d049bb133111eb; return x ^ (x >> 31); } vector subtree_hash, overall_hash; void dfs1(int u, const vector> &adj) { size_t tot = 0; for (int v : adj[u]) { dfs1(v, adj); tot += subtree_hash[v]; } subtree_hash[u] = get_hash(tot); } void dfs2(int u, const vector> &adj, size_t par_hash) { overall_hash[u] = get_hash(subtree_hash[u] + get_hash(par_hash)); for (int v : adj[u]) { dfs2(v, adj, overall_hash[u]); } } int solve() { // Set the root node as a giant square enclosing everything. const LL LO = -5, HI = (int)1e9 + 5; vector> polygons{{{LO, LO}, {HI, LO}, {HI, HI}, {LO, HI}}}; int N, Q; // Input polygons. cin >> N; for (int i = 0, M; i < N; i++) { cin >> M; vector points; for (int j = 0, x, y; j < M; j++) { cin >> x >> y; points.push_back(Point(x, y)); } polygons.push_back(points); } EnclosedPolygons EP(polygons); // Build graph. N++; vector> adj(N); for (int i = 1; i < N; i++) { adj[EP.parent[i]].push_back(i); } subtree_hash.resize(N); overall_hash.resize(N); dfs1(0, adj); dfs2(0, adj, 0); // Process queries. cin >> Q; int ans = 0, E = 0; for (int i = 0, a, b, c, d, e; i < Q; i++) { cin >> a >> b >> c >> d >> e; a ^= E; b ^= E; c ^= E; d ^= E; int u = EP.get_containing_polygon(Point(a, b)); int v = EP.get_containing_polygon(Point(c, d)); if (overall_hash[u] == overall_hash[v]) { ans++; E ^= e; } } return ans; } int main() { ios_base::sync_with_stdio(false); cin.tie(nullptr); int T; cin >> T; for (int t = 1; t <= T; t++) { cout << "Case #" << t << ": " << solve() << endl; } return 0; }