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// 2D segment tree #include "SegTree.h" template< class S, // aggregate data type S (*op) (S, S), // combine aggregate data S (*e) (), // empty element class Coord // for x and y coordinates > struct SegTree2D { using P = pair<Coord, Coord>; // _points must contains all add queries SegTree2D(const vector<P>& _points) : points(_points) { sort(points.begin(), points.end()); points.erase(unique(points.begin(), points.end()), points.end()); n = points.size(); // init segtrees coords.resize(n * 2); for (int i = 0; i < n; i++) { coords[n + i] = {{points[i].second, points[i].first}}; } for (int i = n-1; i > 0; i--) { std::merge(coords[i*2].begin(), coords[i*2].end(), coords[i*2+1].begin(), coords[i*2+1].end(), std::back_inserter(coords[i])); coords[i].erase(unique(coords[i].begin(), coords[i].end()), coords[i].end()); } for (const auto& c : coords) { segs.emplace_back(SegTree<S, op, e>(c.size())); } } // Set value(p) = val void set(P p, S val) { int i = lower_bound(points.begin(), points.end(), p) - points.begin(); assert(i < n && points[i] == p); for (i += n; i; i >>= 1) { int j = lower_bound(coords[i].begin(), coords[i].end(), P{p.second, p.first}) - coords[i].begin(); segs[i].set(j, val); } } // Get value at p S get(P p) const { return prod(p, P{p.first + 1, p.second + 1}); } // Get sum of points in rectangles, given bottom-left and top-right // [low.x, high.x - 1] * [low.y, high.y - 1] S prod(P low, P high) const { assert(low.first <= high.first); assert(low.second <= high.second); if (low.first == high.first) return e(); if (low.second == high.second) return e(); int l = n + (lower_bound(points.begin(), points.end(), low, cmpFirst) - points.begin()); int r = n + (lower_bound(points.begin(), points.end(), high, cmpFirst) - points.begin()); S res = e(); while (l < r) { if (l & 1) res = op(res, prod_1d(l++, low.second, high.second)); if (r & 1) res = op(res, prod_1d(--r, low.second, high.second)); l >>= 1; r >>= 1; } return res; } // private: S prod_1d(int x, Coord l, Coord r) const { auto il = lower_bound(coords[x].begin(), coords[x].end(), P{l, l}, cmpFirst) - coords[x].begin(); auto ir = lower_bound(coords[x].begin(), coords[x].end(), P{r, r}, cmpFirst) - coords[x].begin(); return segs[x].prod(il, ir); } static bool cmpFirst(const P& u, const P& v) { return u.first < v.first; } int n; vector<P> points; // segtrees, outer layer by x-coordinate vector<vector<P>> coords; // coords[i] stores all points maintained by i-th node in ST vector<SegTree<S, op, e>> segs; };
#line 1 "DataStructure/SegTree2D.h" // 2D segment tree #line 1 "DataStructure/SegTree.h" // SegTree, copied from AtCoder library {{{ // AtCoder doc: https://atcoder.github.io/ac-library/master/document_en/segtree.html // // Notes: // - Index of elements from 0 -> n-1 // - Range queries are [l, r-1] // // Tested: // - (binary search) https://atcoder.jp/contests/practice2/tasks/practice2_j // - https://oj.vnoi.info/problem/gss // - https://oj.vnoi.info/problem/nklineup // - (max_right & min_left for delete position queries) https://oj.vnoi.info/problem/segtree_itstr // - https://judge.yosupo.jp/problem/point_add_range_sum // - https://judge.yosupo.jp/problem/point_set_range_composite int ceil_pow2(int n) { int x = 0; while ((1U << x) < (unsigned int)(n)) x++; return x; } template< class T, // data type for nodes T (*op) (T, T), // operator to combine 2 nodes T (*e)() // identity element > struct SegTree { SegTree() : SegTree(0) {} explicit SegTree(int n) : SegTree(vector<T> (n, e())) {} explicit SegTree(const vector<T>& v) : _n((int) v.size()) { log = ceil_pow2(_n); size = 1<<log; d = vector<T> (2*size, e()); for (int i = 0; i < _n; i++) d[size+i] = v[i]; for (int i = size - 1; i >= 1; i--) { update(i); } } // 0 <= p < n void set(int p, T x) { assert(0 <= p && p < _n); p += size; d[p] = x; for (int i = 1; i <= log; i++) update(p >> i); } // 0 <= p < n T get(int p) const { assert(0 <= p && p < _n); return d[p + size]; } // Get product in range [l, r-1] // 0 <= l <= r <= n // For empty segment (l == r) -> return e() T prod(int l, int r) const { assert(0 <= l && l <= r && r <= _n); T sml = e(), smr = e(); l += size; r += size; while (l < r) { if (l & 1) sml = op(sml, d[l++]); if (r & 1) smr = op(d[--r], smr); l >>= 1; r >>= 1; } return op(sml, smr); } T all_prod() const { return d[1]; } // Binary search on SegTree to find largest r: // f(op(a[l] .. a[r-1])) = true (assuming empty array is always true) // f(op(a[l] .. a[r])) = false (assuming op(..., a[n]), which is out of bound, is always false) template <bool (*f)(T)> int max_right(int l) const { return max_right(l, [](T x) { return f(x); }); } template <class F> int max_right(int l, F f) const { assert(0 <= l && l <= _n); assert(f(e())); if (l == _n) return _n; l += size; T sm = e(); do { while (l % 2 == 0) l >>= 1; if (!f(op(sm, d[l]))) { while (l < size) { l = (2 * l); if (f(op(sm, d[l]))) { sm = op(sm, d[l]); l++; } } return l - size; } sm = op(sm, d[l]); l++; } while ((l & -l) != l); return _n; } // Binary search on SegTree to find smallest l: // f(op(a[l] .. a[r-1])) = true (assuming empty array is always true) // f(op(a[l-1] .. a[r-1])) = false (assuming op(a[-1], ..), which is out of bound, is always false) template <bool (*f)(T)> int min_left(int r) const { return min_left(r, [](T x) { return f(x); }); } template <class F> int min_left(int r, F f) const { assert(0 <= r && r <= _n); assert(f(e())); if (r == 0) return 0; r += size; T sm = e(); do { r--; while (r > 1 && (r % 2)) r >>= 1; if (!f(op(d[r], sm))) { while (r < size) { r = (2 * r + 1); if (f(op(d[r], sm))) { sm = op(d[r], sm); r--; } } return r + 1 - size; } sm = op(d[r], sm); } while ((r & -r) != r); return 0; } private: int _n, size, log; vector<T> d; void update(int k) { d[k] = op(d[2*k], d[2*k+1]); } }; // }}} // SegTree examples {{{ // Examples: Commonly used SegTree ops: max / min / sum struct MaxSegTreeOp { static int op(int x, int y) { return max(x, y); } static int e() { return INT_MIN; } }; struct MinSegTreeOp { static int op(int x, int y) { return min(x, y); } static int e() { return INT_MAX; } }; struct SumSegTreeOp { static long long op(long long x, long long y) { return x + y; } static long long e() { return 0; } }; // using STMax = SegTree<int, MaxSegTreeOp::op, MaxSegTreeOp::e>; // using STMin = SegTree<int, MinSegTreeOp::op, MinSegTreeOp::e>; // using STSum = SegTree<int, SumSegTreeOp::op, SumSegTreeOp::e>; // }}} #line 3 "DataStructure/SegTree2D.h" template< class S, // aggregate data type S (*op) (S, S), // combine aggregate data S (*e) (), // empty element class Coord // for x and y coordinates > struct SegTree2D { using P = pair<Coord, Coord>; // _points must contains all add queries SegTree2D(const vector<P>& _points) : points(_points) { sort(points.begin(), points.end()); points.erase(unique(points.begin(), points.end()), points.end()); n = points.size(); // init segtrees coords.resize(n * 2); for (int i = 0; i < n; i++) { coords[n + i] = {{points[i].second, points[i].first}}; } for (int i = n-1; i > 0; i--) { std::merge(coords[i*2].begin(), coords[i*2].end(), coords[i*2+1].begin(), coords[i*2+1].end(), std::back_inserter(coords[i])); coords[i].erase(unique(coords[i].begin(), coords[i].end()), coords[i].end()); } for (const auto& c : coords) { segs.emplace_back(SegTree<S, op, e>(c.size())); } } // Set value(p) = val void set(P p, S val) { int i = lower_bound(points.begin(), points.end(), p) - points.begin(); assert(i < n && points[i] == p); for (i += n; i; i >>= 1) { int j = lower_bound(coords[i].begin(), coords[i].end(), P{p.second, p.first}) - coords[i].begin(); segs[i].set(j, val); } } // Get value at p S get(P p) const { return prod(p, P{p.first + 1, p.second + 1}); } // Get sum of points in rectangles, given bottom-left and top-right // [low.x, high.x - 1] * [low.y, high.y - 1] S prod(P low, P high) const { assert(low.first <= high.first); assert(low.second <= high.second); if (low.first == high.first) return e(); if (low.second == high.second) return e(); int l = n + (lower_bound(points.begin(), points.end(), low, cmpFirst) - points.begin()); int r = n + (lower_bound(points.begin(), points.end(), high, cmpFirst) - points.begin()); S res = e(); while (l < r) { if (l & 1) res = op(res, prod_1d(l++, low.second, high.second)); if (r & 1) res = op(res, prod_1d(--r, low.second, high.second)); l >>= 1; r >>= 1; } return res; } // private: S prod_1d(int x, Coord l, Coord r) const { auto il = lower_bound(coords[x].begin(), coords[x].end(), P{l, l}, cmpFirst) - coords[x].begin(); auto ir = lower_bound(coords[x].begin(), coords[x].end(), P{r, r}, cmpFirst) - coords[x].begin(); return segs[x].prod(il, ir); } static bool cmpFirst(const P& u, const P& v) { return u.first < v.first; } int n; vector<P> points; // segtrees, outer layer by x-coordinate vector<vector<P>> coords; // coords[i] stores all points maintained by i-th node in ST vector<SegTree<S, op, e>> segs; };