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#include "ds/kdtree/kdtree_monoid.hpp"
template <class Monoid, typename XY> struct KDTree_Monoid { using MX = Monoid; using X = typename MX::value_type; static_assert(MX::commute); // 小数も考慮すると、閉で持つ設計方針になる。ただし、クエリはいつもの半開を使う vc<tuple<XY, XY, XY, XY>> closed_range; vc<X> dat; int n; KDTree_Monoid(vc<XY> xs, vc<XY> ys, vc<X> vs) : n(len(xs)) { assert(n > 0); int log = 0; while ((1 << log) < n) ++log; dat.resize(1 << (log + 1)); closed_range.resize(1 << (log + 1)); build(1, xs, ys, vs); } void multiply(XY x, XY y, const X& v) { multiply_rec(1, x, y, v); } // [xl, xr) x [yl, yr) X prod(XY xl, XY xr, XY yl, XY yr) { assert(xl <= xr && yl <= yr); return prod_rec(1, xl, xr, yl, yr); } X prod_all() { return dat[1]; } private: void build(int idx, vc<XY> xs, vc<XY> ys, vc<X> vs, bool divx = true) { int n = len(xs); auto& [xmin, xmax, ymin, ymax] = closed_range[idx]; xmin = ymin = infty<XY>; xmax = ymax = -infty<XY>; FOR(i, n) { auto x = xs[i], y = ys[i]; chmin(xmin, x), chmax(xmax, x), chmin(ymin, y), chmax(ymax, y); } if (xmin == xmax && ymin == ymax) { X x = MX::unit(); for (auto&& v: vs) x = MX::op(x, v); dat[idx] = x; return; } int m = n / 2; vc<int> I(n); iota(all(I), 0); if (divx) { nth_element(I.begin(), I.begin() + m, I.end(), [xs](int i, int j) { return xs[i] < xs[j]; }); } else { nth_element(I.begin(), I.begin() + m, I.end(), [ys](int i, int j) { return ys[i] < ys[j]; }); } xs = rearrange(xs, I), ys = rearrange(ys, I), vs = rearrange(vs, I); build(2 * idx + 0, {xs.begin(), xs.begin() + m}, {ys.begin(), ys.begin() + m}, {vs.begin(), vs.begin() + m}, !divx); build(2 * idx + 1, {xs.begin() + m, xs.end()}, {ys.begin() + m, ys.end()}, {vs.begin() + m, vs.end()}, !divx); dat[idx] = MX::op(dat[2 * idx + 0], dat[2 * idx + 1]); } inline bool is_leaf(int idx) { auto& [xmin, xmax, ymin, ymax] = closed_range[idx]; return xmin == xmax && ymin == ymax; } inline bool isin(XY x, XY y, int idx) { auto& [xmin, xmax, ymin, ymax] = closed_range[idx]; return (xmin <= x && x <= xmax && ymin <= y && y <= ymax); } bool multiply_rec(int idx, XY x, XY y, X v) { if (!isin(x, y, idx)) return false; if (is_leaf(idx)) { dat[idx] = MX::op(dat[idx], v); return true; } bool done = 0; if (multiply_rec(2 * idx + 0, x, y, v)) done = 1; if (!done && multiply_rec(2 * idx + 1, x, y, v)) done = 1; if (done) { dat[idx] = MX::op(dat[2 * idx + 0], dat[2 * idx + 1]); } return done; } X prod_rec(int idx, XY x1, XY x2, XY y1, XY y2) { auto& [xmin, xmax, ymin, ymax] = closed_range[idx]; if (x2 <= xmin || xmax < x1) return MX::unit(); if (y2 <= ymin || ymax < y1) return MX::unit(); if (x1 <= xmin && xmax < x2 && y1 <= ymin && ymax < y2) { return dat[idx]; } return MX::op(prod_rec(2 * idx + 0, x1, x2, y1, y2), prod_rec(2 * idx + 1, x1, x2, y1, y2)); } };
#line 1 "ds/kdtree/kdtree_monoid.hpp" template <class Monoid, typename XY> struct KDTree_Monoid { using MX = Monoid; using X = typename MX::value_type; static_assert(MX::commute); // 小数も考慮すると、閉で持つ設計方針になる。ただし、クエリはいつもの半開を使う vc<tuple<XY, XY, XY, XY>> closed_range; vc<X> dat; int n; KDTree_Monoid(vc<XY> xs, vc<XY> ys, vc<X> vs) : n(len(xs)) { assert(n > 0); int log = 0; while ((1 << log) < n) ++log; dat.resize(1 << (log + 1)); closed_range.resize(1 << (log + 1)); build(1, xs, ys, vs); } void multiply(XY x, XY y, const X& v) { multiply_rec(1, x, y, v); } // [xl, xr) x [yl, yr) X prod(XY xl, XY xr, XY yl, XY yr) { assert(xl <= xr && yl <= yr); return prod_rec(1, xl, xr, yl, yr); } X prod_all() { return dat[1]; } private: void build(int idx, vc<XY> xs, vc<XY> ys, vc<X> vs, bool divx = true) { int n = len(xs); auto& [xmin, xmax, ymin, ymax] = closed_range[idx]; xmin = ymin = infty<XY>; xmax = ymax = -infty<XY>; FOR(i, n) { auto x = xs[i], y = ys[i]; chmin(xmin, x), chmax(xmax, x), chmin(ymin, y), chmax(ymax, y); } if (xmin == xmax && ymin == ymax) { X x = MX::unit(); for (auto&& v: vs) x = MX::op(x, v); dat[idx] = x; return; } int m = n / 2; vc<int> I(n); iota(all(I), 0); if (divx) { nth_element(I.begin(), I.begin() + m, I.end(), [xs](int i, int j) { return xs[i] < xs[j]; }); } else { nth_element(I.begin(), I.begin() + m, I.end(), [ys](int i, int j) { return ys[i] < ys[j]; }); } xs = rearrange(xs, I), ys = rearrange(ys, I), vs = rearrange(vs, I); build(2 * idx + 0, {xs.begin(), xs.begin() + m}, {ys.begin(), ys.begin() + m}, {vs.begin(), vs.begin() + m}, !divx); build(2 * idx + 1, {xs.begin() + m, xs.end()}, {ys.begin() + m, ys.end()}, {vs.begin() + m, vs.end()}, !divx); dat[idx] = MX::op(dat[2 * idx + 0], dat[2 * idx + 1]); } inline bool is_leaf(int idx) { auto& [xmin, xmax, ymin, ymax] = closed_range[idx]; return xmin == xmax && ymin == ymax; } inline bool isin(XY x, XY y, int idx) { auto& [xmin, xmax, ymin, ymax] = closed_range[idx]; return (xmin <= x && x <= xmax && ymin <= y && y <= ymax); } bool multiply_rec(int idx, XY x, XY y, X v) { if (!isin(x, y, idx)) return false; if (is_leaf(idx)) { dat[idx] = MX::op(dat[idx], v); return true; } bool done = 0; if (multiply_rec(2 * idx + 0, x, y, v)) done = 1; if (!done && multiply_rec(2 * idx + 1, x, y, v)) done = 1; if (done) { dat[idx] = MX::op(dat[2 * idx + 0], dat[2 * idx + 1]); } return done; } X prod_rec(int idx, XY x1, XY x2, XY y1, XY y2) { auto& [xmin, xmax, ymin, ymax] = closed_range[idx]; if (x2 <= xmin || xmax < x1) return MX::unit(); if (y2 <= ymin || ymax < y1) return MX::unit(); if (x1 <= xmin && xmax < x2 && y1 <= ymin && ymax < y2) { return dat[idx]; } return MX::op(prod_rec(2 * idx + 0, x1, x2, y1, y2), prod_rec(2 * idx + 1, x1, x2, y1, y2)); } };