Segment-Tree(2次元) (src/DataStructure/SegmentTree_2D.hpp)

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Last update: 2024-09-28 21:50:54+09:00
Include: #include "src/DataStructure/SegmentTree_2D.hpp"

半群は乗らない. モノイドの単位元 M::ti は明示的に与える必要あり.

テンプレート `SegmentTree_2D<class pos_t, class M>`

pos_t : 座標(一次元)を表現する型. 通常 int とか使う.
M : いつものモノイドのやつを与える.

メンバ関数

使用例で出てくるRMQは次のような感じ.

struct RMQ{
 using T=int;
 static T ti(){return 1<<31;}
 static T op(T l,T r){return min(l,r);}
};

名前	概要
`SegmentTree_2D(P* p, n)` `SegmentTree_2D(vector<P>& p)` `SegmentTree_2D(set<P>& p)` (クラス`P` は `tuple<pos_t,pos_t>` like)	コンストラクタ. n個の点の座標を与える. この時の初期値はモノイドの単位元.

使用例

vector<array<int>> xy(N);
for(auto&[x,y]:xy) cin>>x>>y;
SegmentTree_2D<int,RMQ> seg(xy);


`SegmentTree_2D(P* p,int n, U v)` `SegmentTree_2D(vector<P>& p, U v)` `SegmentTree_2D(set<P>& p, U v)` (クラス`P` は `tuple<pos_t,pos_t>` like)	コンストラクタ. n個の点の座標と点に乗せる共通の初期値を与える.

使用例

vector<array<int>> xy(N);
for(auto&[x,y]:xy) cin>>x>>y;
SegmentTree_2D<int,RMQ> seg(xy,1);


`SegmentTree_2D(P* p,int n)` `SegmentTree_2D(vector<P>& p)` (クラス`P` は `tuple<pos_t,pos_t,T>` like)	コンストラクタ. n個の点の座標と各々の点に乗せる初期値を与える.

使用例

vector<array<int,3>> xyv(N);
for(auto&[x,y,v]:xyv) cin>>x>>y>>v;
SegmentTree_2D<int,RMQ> seg(xyv);


`SegmentTree_2D(pair<P,U>* p,int n)` `SegmentTree_2D(vector<pair<P,U>>& p)` `SegmentTree_2D(map<P,U>& p)` (クラス`P` は `tuple<pos_t,pos_t>` like)	コンストラクタ. n個の点の座標と各々の点に乗せる初期値を与える.

使用例

map<array<int>,int> xyv;
for(int i=0;i<N;++i){
 int x,y,v;cin>>x>>y>>v;
 xyv[{x,y}] += v;
}
SegmentTree_2D<int,RMQ> seg(xyv);

		計算量
`prod(l,r,u,d)`	直方体(長方形) 内部に位置する点についてその点に乗っている値を集約した値を返す. ※半開区間 $\lbrack l,r )\times \lbrack u,d)$	以下, 点の個数を $n$ とする. $O((\log n)^2)$
`set(x,y,v)`	点 $(x,y)$ の値を `v` に変更する. 点が存在しないとassertで落ちる.	$O(\log n)$
`get(x,y)`	点 $(x,y)$ の値を返す.　点が存在しないとassertで落ちる.	$O(\log n)$
`mul(x,y,v)`	点 $(x,y)$ の値に `v` を (モノイド演算で) かける. 点が存在しないとassertで落ちる.	$O(\log n)$

Verify

技術室奥プログラミングコンテスト#6 Day1 N - Jump and Walk (2次元 min, kdtだとTLE)

Depends on

tupleやarrayに関するテンプレート他 (src/Internal/tuple_traits.hpp)

Verified with

Code

#pragma once
#include <vector>
#include <algorithm>
#include <numeric>
#include <map>
#include <set>
#include <cassert>
#include "src/Internal/tuple_traits.hpp"
template <class pos_t, class M> class SegmentTree_2D {
 using T= typename M::T;
 using Pos= std::array<pos_t, 2>;
 std::vector<pos_t> xs;
 std::vector<Pos> yxs;
 std::vector<int> id, tol;
 std::vector<T> val;
 template <class P> using canbe_Pos= std::is_convertible<to_tuple_t<P>, std::tuple<pos_t, pos_t>>;
 template <class P> using canbe_PosV= std::is_convertible<to_tuple_t<P>, std::tuple<pos_t, pos_t, T>>;
 template <class P, class U> static constexpr bool canbe_Pos_and_T_v= std::conjunction_v<canbe_Pos<P>, std::is_convertible<U, T>>;
 int sz;
 inline int x2i(pos_t x) const { return std::lower_bound(xs.begin(), xs.end(), x) - xs.begin(); }
 inline int y2i(pos_t y) const {
  return std::lower_bound(yxs.begin(), yxs.end(), Pos{y, 0}, [](const Pos &a, const Pos &b) { return a[0] < b[0]; }) - yxs.begin();
 }
 inline int xy2i(pos_t x, pos_t y) const {
  Pos p{y, x};
  auto it= std::lower_bound(yxs.begin(), yxs.end(), p);
  return assert(p == *it), it - yxs.begin();
 }
 template <bool z, size_t k, class P> inline auto get_(const P &p) {
  if constexpr (z) return std::get<k>(p);
  else return std::get<k>(p.first);
 }
 template <bool z, class XYW> inline void build(const XYW *xyw, int n, const T &v= M::ti()) {
  xs.resize(n);
  for (int i= n; i--;) xs[i]= get_<z, 0>(xyw[i]);
  std::sort(xs.begin(), xs.end()), xs.erase(std::unique(xs.begin(), xs.end()), xs.end()), id.resize((sz= 1 << (32 - __builtin_clz(xs.size()))) + xs.size() + 1);
  std::vector<int> ord(n);
  for (int j= n; j--;)
   for (int i= x2i(get_<z, 0>(xyw[j])) + sz; i; i>>= 1) ++id[i + 1];
  for (int i= 1, e= sz + xs.size(); i < e; ++i) id[i + 1]+= id[i];
  val.assign(id.back() * 2, M::ti()), tol.resize(id[sz] + 1), std::iota(ord.begin(), ord.end(), 0), std::sort(ord.begin(), ord.end(), [&](int i, int j) { return get_<z, 1>(xyw[i]) == get_<z, 1>(xyw[j]) ? get_<z, 0>(xyw[i]) < get_<z, 0>(xyw[j]) : get_<z, 1>(xyw[i]) < get_<z, 1>(xyw[j]); });
  {
   std::vector<int> ptr= id;
   for (int r: ord)
    for (int i= x2i(get_<z, 0>(xyw[r])) + sz, j= -1; i; j= i, i>>= 1) {
     int p= ptr[i]++;
     if constexpr (z) {
      if constexpr (std::tuple_size_v<XYW> == 3) val[id[i + 1] + p]= std::get<2>(xyw[r]);
      else val[id[i + 1] + p]= v;
     } else val[id[i + 1] + p]= xyw[r].second;
     if (j != -1) tol[p + 1]= !(j & 1);
    }
   for (int i= 1, e= id[sz]; i < e; ++i) tol[i + 1]+= tol[i];
   for (int i= 0, e= sz + xs.size(); i < e; ++i) {
    auto dat= val.begin() + id[i] * 2;
    for (int j= id[i + 1] - id[i]; --j > 0;) dat[j]= M::op(dat[j * 2], dat[j * 2 + 1]);
   }
  }
  yxs.resize(n);
  for (int i= n; i--;) yxs[i]= {get_<z, 1>(xyw[ord[i]]), get_<z, 0>(xyw[ord[i]])};
 }
 inline T prdi(int i, int a, int b) const {
  int n= id[i + 1] - id[i];
  T ret= M::ti();
  auto dat= val.begin() + id[i] * 2;
  for (a+= n, b+= n; a < b; a>>= 1, b>>= 1) {
   if (a & 1) ret= M::op(ret, dat[a++]);
   if (b & 1) ret= M::op(dat[--b], ret);
  }
  return ret;
 }
 template <bool z> inline void seti(int i, int j, T v) {
  auto dat= val.begin() + id[i] * 2;
  j+= id[i + 1] - id[i];
  if constexpr (z) dat[j]= v;
  else dat[j]= M::op(dat[j], v);
  for (; j;) j>>= 1, dat[j]= M::op(dat[2 * j], dat[2 * j + 1]);
 }
 template <bool z> inline void set_(pos_t x, pos_t y, T v) {
  for (int i= 1, p= xy2i(x, y);;) {
   if (seti<z>(i, p - id[i], v); i >= sz) break;
   if (int lc= tol[p] - tol[id[i]], rc= (p - id[i]) - lc; tol[p + 1] - tol[p]) p= id[2 * i] + lc, i= 2 * i;
   else p= id[2 * i + 1] + rc, i= 2 * i + 1;
  }
 }
public:
 template <class P, typename= std::enable_if_t<std::disjunction_v<canbe_Pos<P>, canbe_PosV<P>>>> SegmentTree_2D(const P *p, size_t n) { build<1>(p, n); }
 template <class P, typename= std::enable_if_t<std::disjunction_v<canbe_Pos<P>, canbe_PosV<P>>>> SegmentTree_2D(const std::vector<P> &p): SegmentTree_2D(p.data(), p.size()) {}
 template <class P, typename= std::enable_if_t<canbe_Pos<P>::value>> SegmentTree_2D(const std::set<P> &p): SegmentTree_2D(std::vector(p.begin(), p.end())) {}
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const P *p, size_t n, const U &v) { build<1>(p, n, v); }
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const std::vector<P> &p, const U &v): SegmentTree_2D(p.data(), p.size(), v) {}
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const std::set<P> &p, const U &v): SegmentTree_2D(std::vector(p.begin(), p.end()), v) {}
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const std::pair<P, U> *p, size_t n) { build<0>(p, n); }
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const std::vector<std::pair<P, U>> &p): SegmentTree_2D(p.data(), p.size()) {}
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const std::map<P, U> &p): SegmentTree_2D(std::vector(p.begin(), p.end())) {}
 // [l,r) x [u,d)
 T prod(pos_t l, pos_t r, pos_t u, pos_t d) const {
  T ret= M::ti();
  int L= x2i(l), R= x2i(r);
  auto dfs= [&](auto &dfs, int i, int a, int b, int c, int d) -> void {
   if (c == d || R <= a || b <= L) return;
   if (L <= a && b <= R) return ret= M::op(ret, prdi(i, c, d)), void();
   int m= (a + b) / 2, ac= tol[id[i] + c] - tol[id[i]], bc= c - ac, ad= tol[id[i] + d] - tol[id[i]], bd= d - ad;
   dfs(dfs, i * 2, a, m, ac, ad), dfs(dfs, i * 2 + 1, m, b, bc, bd);
  };
  return dfs(dfs, 1, 0, sz, y2i(u), y2i(d)), ret;
 }
 void set(pos_t x, pos_t y, T v) { set_<1>(x, y, v); }
 void mul(pos_t x, pos_t y, T v) { set_<0>(x, y, v); }
 T get(pos_t x, pos_t y) const { return val[xy2i(x, y) + id[2]]; }
};

#line 2 "src/DataStructure/SegmentTree_2D.hpp"
#include <vector>
#include <algorithm>
#include <numeric>
#include <map>
#include <set>
#include <cassert>
#line 2 "src/Internal/tuple_traits.hpp"
#include <tuple>
#include <array>
#include <type_traits>
#include <cstddef>
template <class T> static constexpr bool tuple_like_v= false;
template <class... Args> static constexpr bool tuple_like_v<std::tuple<Args...>> = true;
template <class T, class U> static constexpr bool tuple_like_v<std::pair<T, U>> = true;
template <class T, size_t K> static constexpr bool tuple_like_v<std::array<T, K>> = true;
template <class T> auto to_tuple(const T &t) {
 if constexpr (tuple_like_v<T>) return std::apply([](auto &&...x) { return std::make_tuple(x...); }, t);
}
template <class T> auto forward_tuple(const T &t) {
 if constexpr (tuple_like_v<T>) return std::apply([](auto &&...x) { return std::forward_as_tuple(x...); }, t);
}
template <class T> static constexpr bool array_like_v= false;
template <class T, size_t K> static constexpr bool array_like_v<std::array<T, K>> = true;
template <class T, class U> static constexpr bool array_like_v<std::pair<T, U>> = std::is_convertible_v<T, U>;
template <class T> static constexpr bool array_like_v<std::tuple<T>> = true;
template <class T, class U, class... Args> static constexpr bool array_like_v<std::tuple<T, U, Args...>> = array_like_v<std::tuple<T, Args...>> && std::is_convertible_v<U, T>;
template <class T> auto to_array(const T &t) {
 if constexpr (array_like_v<T>) return std::apply([](auto &&...x) { return std::array{x...}; }, t);
}
template <class T> using to_tuple_t= decltype(to_tuple(T()));
template <class T> using to_array_t= decltype(to_array(T()));
#line 9 "src/DataStructure/SegmentTree_2D.hpp"
template <class pos_t, class M> class SegmentTree_2D {
 using T= typename M::T;
 using Pos= std::array<pos_t, 2>;
 std::vector<pos_t> xs;
 std::vector<Pos> yxs;
 std::vector<int> id, tol;
 std::vector<T> val;
 template <class P> using canbe_Pos= std::is_convertible<to_tuple_t<P>, std::tuple<pos_t, pos_t>>;
 template <class P> using canbe_PosV= std::is_convertible<to_tuple_t<P>, std::tuple<pos_t, pos_t, T>>;
 template <class P, class U> static constexpr bool canbe_Pos_and_T_v= std::conjunction_v<canbe_Pos<P>, std::is_convertible<U, T>>;
 int sz;
 inline int x2i(pos_t x) const { return std::lower_bound(xs.begin(), xs.end(), x) - xs.begin(); }
 inline int y2i(pos_t y) const {
  return std::lower_bound(yxs.begin(), yxs.end(), Pos{y, 0}, [](const Pos &a, const Pos &b) { return a[0] < b[0]; }) - yxs.begin();
 }
 inline int xy2i(pos_t x, pos_t y) const {
  Pos p{y, x};
  auto it= std::lower_bound(yxs.begin(), yxs.end(), p);
  return assert(p == *it), it - yxs.begin();
 }
 template <bool z, size_t k, class P> inline auto get_(const P &p) {
  if constexpr (z) return std::get<k>(p);
  else return std::get<k>(p.first);
 }
 template <bool z, class XYW> inline void build(const XYW *xyw, int n, const T &v= M::ti()) {
  xs.resize(n);
  for (int i= n; i--;) xs[i]= get_<z, 0>(xyw[i]);
  std::sort(xs.begin(), xs.end()), xs.erase(std::unique(xs.begin(), xs.end()), xs.end()), id.resize((sz= 1 << (32 - __builtin_clz(xs.size()))) + xs.size() + 1);
  std::vector<int> ord(n);
  for (int j= n; j--;)
   for (int i= x2i(get_<z, 0>(xyw[j])) + sz; i; i>>= 1) ++id[i + 1];
  for (int i= 1, e= sz + xs.size(); i < e; ++i) id[i + 1]+= id[i];
  val.assign(id.back() * 2, M::ti()), tol.resize(id[sz] + 1), std::iota(ord.begin(), ord.end(), 0), std::sort(ord.begin(), ord.end(), [&](int i, int j) { return get_<z, 1>(xyw[i]) == get_<z, 1>(xyw[j]) ? get_<z, 0>(xyw[i]) < get_<z, 0>(xyw[j]) : get_<z, 1>(xyw[i]) < get_<z, 1>(xyw[j]); });
  {
   std::vector<int> ptr= id;
   for (int r: ord)
    for (int i= x2i(get_<z, 0>(xyw[r])) + sz, j= -1; i; j= i, i>>= 1) {
     int p= ptr[i]++;
     if constexpr (z) {
      if constexpr (std::tuple_size_v<XYW> == 3) val[id[i + 1] + p]= std::get<2>(xyw[r]);
      else val[id[i + 1] + p]= v;
     } else val[id[i + 1] + p]= xyw[r].second;
     if (j != -1) tol[p + 1]= !(j & 1);
    }
   for (int i= 1, e= id[sz]; i < e; ++i) tol[i + 1]+= tol[i];
   for (int i= 0, e= sz + xs.size(); i < e; ++i) {
    auto dat= val.begin() + id[i] * 2;
    for (int j= id[i + 1] - id[i]; --j > 0;) dat[j]= M::op(dat[j * 2], dat[j * 2 + 1]);
   }
  }
  yxs.resize(n);
  for (int i= n; i--;) yxs[i]= {get_<z, 1>(xyw[ord[i]]), get_<z, 0>(xyw[ord[i]])};
 }
 inline T prdi(int i, int a, int b) const {
  int n= id[i + 1] - id[i];
  T ret= M::ti();
  auto dat= val.begin() + id[i] * 2;
  for (a+= n, b+= n; a < b; a>>= 1, b>>= 1) {
   if (a & 1) ret= M::op(ret, dat[a++]);
   if (b & 1) ret= M::op(dat[--b], ret);
  }
  return ret;
 }
 template <bool z> inline void seti(int i, int j, T v) {
  auto dat= val.begin() + id[i] * 2;
  j+= id[i + 1] - id[i];
  if constexpr (z) dat[j]= v;
  else dat[j]= M::op(dat[j], v);
  for (; j;) j>>= 1, dat[j]= M::op(dat[2 * j], dat[2 * j + 1]);
 }
 template <bool z> inline void set_(pos_t x, pos_t y, T v) {
  for (int i= 1, p= xy2i(x, y);;) {
   if (seti<z>(i, p - id[i], v); i >= sz) break;
   if (int lc= tol[p] - tol[id[i]], rc= (p - id[i]) - lc; tol[p + 1] - tol[p]) p= id[2 * i] + lc, i= 2 * i;
   else p= id[2 * i + 1] + rc, i= 2 * i + 1;
  }
 }
public:
 template <class P, typename= std::enable_if_t<std::disjunction_v<canbe_Pos<P>, canbe_PosV<P>>>> SegmentTree_2D(const P *p, size_t n) { build<1>(p, n); }
 template <class P, typename= std::enable_if_t<std::disjunction_v<canbe_Pos<P>, canbe_PosV<P>>>> SegmentTree_2D(const std::vector<P> &p): SegmentTree_2D(p.data(), p.size()) {}
 template <class P, typename= std::enable_if_t<canbe_Pos<P>::value>> SegmentTree_2D(const std::set<P> &p): SegmentTree_2D(std::vector(p.begin(), p.end())) {}
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const P *p, size_t n, const U &v) { build<1>(p, n, v); }
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const std::vector<P> &p, const U &v): SegmentTree_2D(p.data(), p.size(), v) {}
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const std::set<P> &p, const U &v): SegmentTree_2D(std::vector(p.begin(), p.end()), v) {}
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const std::pair<P, U> *p, size_t n) { build<0>(p, n); }
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const std::vector<std::pair<P, U>> &p): SegmentTree_2D(p.data(), p.size()) {}
 template <class P, class U, typename= std::enable_if_t<canbe_Pos_and_T_v<P, U>>> SegmentTree_2D(const std::map<P, U> &p): SegmentTree_2D(std::vector(p.begin(), p.end())) {}
 // [l,r) x [u,d)
 T prod(pos_t l, pos_t r, pos_t u, pos_t d) const {
  T ret= M::ti();
  int L= x2i(l), R= x2i(r);
  auto dfs= [&](auto &dfs, int i, int a, int b, int c, int d) -> void {
   if (c == d || R <= a || b <= L) return;
   if (L <= a && b <= R) return ret= M::op(ret, prdi(i, c, d)), void();
   int m= (a + b) / 2, ac= tol[id[i] + c] - tol[id[i]], bc= c - ac, ad= tol[id[i] + d] - tol[id[i]], bd= d - ad;
   dfs(dfs, i * 2, a, m, ac, ad), dfs(dfs, i * 2 + 1, m, b, bc, bd);
  };
  return dfs(dfs, 1, 0, sz, y2i(u), y2i(d)), ret;
 }
 void set(pos_t x, pos_t y, T v) { set_<1>(x, y, v); }
 void mul(pos_t x, pos_t y, T v) { set_<0>(x, y, v); }
 T get(pos_t x, pos_t y) const { return val[xy2i(x, y) + id[2]]; }
};

Segment-Tree(2次元) (src/DataStructure/SegmentTree_2D.hpp)

テンプレート SegmentTree_2D<class pos_t, class M>

メンバ関数

Verify

Depends on

Verified with

Code

テンプレート `SegmentTree_2D<class pos_t, class M>`