Static Rectangle Add Rectangle Sum (other/static-rectangle-add-rectangle-sum.hpp)

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Last update: 2024-08-10 03:11:07+09:00
Include: #include "other/static-rectangle-add-rectangle-sum.hpp"

2 次元平面があります。

最初に、長方形内のすべての点に重みを加算するクエリが与えられます。

その後、長方形内に含まれる点の重みの総和を求めるいくつかのクエリに答えます。

コンストラクタ

(1) StaticRectangleAddRectangleSum< T, C >()
(2) StaticRectangleAddRectangleSum< T, C >(int n, int q)

T は座標が収まる型、C は重みの総和が収まる型を指定してください。

(2) で長方形の個数 $n$、クエリの個数 $q$ を指定した場合、領域を reserve するので少しだけ効率的です。

add_rectangle

void add_rectangle(T l, T d, T r, T u, C w)

$\lbrace (x,y):l \leq x \lt r, d \leq y \lt u\rbrace$ で表される長方形内にある点に重み $w$ を加算します。

制約

$l \lt r$
$d \lt u$

計算量

$O(1)$

add_query

void add_query(T l, T d, T r, T u)

$\lbrace (x,y):l \leq x \lt r, d \leq y \lt u\rbrace$ で表される長方形内にある点の重みの総和を求めるクエリを追加します。

制約

$l \lt r$
$d \lt u$

計算量

$O(1)$

calculate_queries

vector<C> calculate_queries() const

それぞれのクエリの答えを返します。$i$ 番目の要素は $i$ 番目に追加したクエリの答えが格納されます。

計算量

$O((n + q) \log (n + q))$

Depends on

Binary Indexed Tree (structure/others/binary-indexed-tree.hpp)

Verified with

test/verify/yosupo-static-rectangle-add-rectangle-sum.test.cpp

Code

#include "../structure/others/binary-indexed-tree.hpp"

template <typename T, typename C>
struct StaticRectangleAddRectangleSum {
  struct Hikari : array<C, 4> {
    Hikari &operator+=(const Hikari &p) {
      for (int i = 0; i < 4; i++) {
        this->at(i) += p.at(i);
      }
      return *this;
    }
  };

  using BIT = BinaryIndexedTree<Hikari>;

  static_assert(is_integral<T>::value,
                "template parameter T must be integral type");

  struct Rectangle {
    T l, d, r, u;
    C w;
  };

  struct Query {
    T l, d, r, u;
  };

  vector<Rectangle> rectangles;
  vector<Query> queries;

  StaticRectangleAddRectangleSum() = default;

  StaticRectangleAddRectangleSum(int n, int q) {
    rectangles.reserve(n);
    queries.reserve(q);
  }

  void add_rectangle(T l, T d, T r, T u, C w) {
    rectangles.emplace_back(Rectangle{l, d, r, u, w});
  }

  // total weight of [l, r) * [d, u) points
  void add_query(T l, T d, T r, T u) {
    queries.emplace_back(Query{l, d, r, u});
  }

  vector<C> calculate_queries() {
    int n = (int)rectangles.size();
    int q = (int)queries.size();
    vector<C> ans(q);
    if (rectangles.empty() or queries.empty()) {
      return ans;
    }
    vector<T> ys;
    ys.reserve(n + n);
    for (Rectangle &p : rectangles) {
      ys.emplace_back(p.d);
      ys.emplace_back(p.u);
    }
    sort(ys.begin(), ys.end());
    ys.erase(unique(ys.begin(), ys.end()), ys.end());

    struct Q {
      T x;
      int d, u;
      bool type;
      int idx;
    };
    vector<Q> rs, qs;
    rs.reserve(n + n);
    qs.reserve(q + q);
    for (int i = 0; i < n; i++) {
      auto &p = rectangles[i];
      int d = lower_bound(ys.begin(), ys.end(), p.d) - ys.begin();
      int u = lower_bound(ys.begin(), ys.end(), p.u) - ys.begin();
      rs.emplace_back(Q{p.l, d, u, false, i});
      rs.emplace_back(Q{p.r, d, u, true, i});
    }
    for (int i = 0; i < q; i++) {
      auto &p = queries[i];
      int d = lower_bound(ys.begin(), ys.end(), p.d) - ys.begin();
      int u = lower_bound(ys.begin(), ys.end(), p.u) - ys.begin();
      qs.emplace_back(Q{p.l, d, u, false, i});
      qs.emplace_back(Q{p.r, d, u, true, i});
    }
    sort(rs.begin(), rs.end(),
         [](const Q &a, const Q &b) { return a.x < b.x; });
    sort(qs.begin(), qs.end(),
         [](const Q &a, const Q &b) { return a.x < b.x; });
    int j = 0;
    BIT bit(ys.size());
    for (auto &query : qs) {
      while (j < n + n and rs[j].x < query.x) {
        auto &p = rectangles[rs[j].idx];
        if (rs[j].type) {
          bit.apply(rs[j].d, {-p.w * p.r * p.d, -p.w, p.w * p.d, p.w * p.r});
          bit.apply(rs[j].u, {p.w * p.r * p.u, p.w, -p.w * p.u, -p.w * p.r});
        } else {
          bit.apply(rs[j].d, {p.w * p.l * p.d, p.w, -p.w * p.d, -p.w * p.l});
          bit.apply(rs[j].u, {-p.w * p.l * p.u, -p.w, p.w * p.u, p.w * p.l});
        }
        ++j;
      }
      auto &p = queries[query.idx];
      auto uret = bit.prod(query.u);
      ans[query.idx] += uret[0];
      ans[query.idx] += uret[1] * query.x * p.u;
      ans[query.idx] += uret[2] * query.x;
      ans[query.idx] += uret[3] * p.u;
      auto dret = bit.prod(query.d);
      ans[query.idx] -= dret[0];
      ans[query.idx] -= dret[1] * query.x * p.d;
      ans[query.idx] -= dret[2] * query.x;
      ans[query.idx] -= dret[3] * p.d;
      if (not query.type) ans[query.idx] *= -1;
    }
    return ans;
  }
};

#line 1 "structure/others/binary-indexed-tree.hpp"
template <typename T>
struct BinaryIndexedTree {
 private:
  int n;
  vector<T> data;

 public:
  BinaryIndexedTree() = default;

  explicit BinaryIndexedTree(int n) : n(n) { data.assign(n + 1, T()); }

  explicit BinaryIndexedTree(const vector<T> &v)
      : BinaryIndexedTree((int)v.size()) {
    build(v);
  }

  void build(const vector<T> &v) {
    assert(n == (int)v.size());
    for (int i = 1; i <= n; i++) data[i] = v[i - 1];
    for (int i = 1; i <= n; i++) {
      int j = i + (i & -i);
      if (j <= n) data[j] += data[i];
    }
  }

  void apply(int k, const T &x) {
    for (++k; k <= n; k += k & -k) data[k] += x;
  }

  T prod(int r) const {
    T ret = T();
    for (; r > 0; r -= r & -r) ret += data[r];
    return ret;
  }

  T prod(int l, int r) const { return prod(r) - prod(l); }

  int lower_bound(T x) const {
    int i = 0;
    for (int k = 1 << (__lg(n) + 1); k > 0; k >>= 1) {
      if (i + k <= n && data[i + k] < x) {
        x -= data[i + k];
        i += k;
      }
    }
    return i;
  }

  int upper_bound(T x) const {
    int i = 0;
    for (int k = 1 << (__lg(n) + 1); k > 0; k >>= 1) {
      if (i + k <= n && data[i + k] <= x) {
        x -= data[i + k];
        i += k;
      }
    }
    return i;
  }
};
#line 2 "other/static-rectangle-add-rectangle-sum.hpp"

template <typename T, typename C>
struct StaticRectangleAddRectangleSum {
  struct Hikari : array<C, 4> {
    Hikari &operator+=(const Hikari &p) {
      for (int i = 0; i < 4; i++) {
        this->at(i) += p.at(i);
      }
      return *this;
    }
  };

  using BIT = BinaryIndexedTree<Hikari>;

  static_assert(is_integral<T>::value,
                "template parameter T must be integral type");

  struct Rectangle {
    T l, d, r, u;
    C w;
  };

  struct Query {
    T l, d, r, u;
  };

  vector<Rectangle> rectangles;
  vector<Query> queries;

  StaticRectangleAddRectangleSum() = default;

  StaticRectangleAddRectangleSum(int n, int q) {
    rectangles.reserve(n);
    queries.reserve(q);
  }

  void add_rectangle(T l, T d, T r, T u, C w) {
    rectangles.emplace_back(Rectangle{l, d, r, u, w});
  }

  // total weight of [l, r) * [d, u) points
  void add_query(T l, T d, T r, T u) {
    queries.emplace_back(Query{l, d, r, u});
  }

  vector<C> calculate_queries() {
    int n = (int)rectangles.size();
    int q = (int)queries.size();
    vector<C> ans(q);
    if (rectangles.empty() or queries.empty()) {
      return ans;
    }
    vector<T> ys;
    ys.reserve(n + n);
    for (Rectangle &p : rectangles) {
      ys.emplace_back(p.d);
      ys.emplace_back(p.u);
    }
    sort(ys.begin(), ys.end());
    ys.erase(unique(ys.begin(), ys.end()), ys.end());

    struct Q {
      T x;
      int d, u;
      bool type;
      int idx;
    };
    vector<Q> rs, qs;
    rs.reserve(n + n);
    qs.reserve(q + q);
    for (int i = 0; i < n; i++) {
      auto &p = rectangles[i];
      int d = lower_bound(ys.begin(), ys.end(), p.d) - ys.begin();
      int u = lower_bound(ys.begin(), ys.end(), p.u) - ys.begin();
      rs.emplace_back(Q{p.l, d, u, false, i});
      rs.emplace_back(Q{p.r, d, u, true, i});
    }
    for (int i = 0; i < q; i++) {
      auto &p = queries[i];
      int d = lower_bound(ys.begin(), ys.end(), p.d) - ys.begin();
      int u = lower_bound(ys.begin(), ys.end(), p.u) - ys.begin();
      qs.emplace_back(Q{p.l, d, u, false, i});
      qs.emplace_back(Q{p.r, d, u, true, i});
    }
    sort(rs.begin(), rs.end(),
         [](const Q &a, const Q &b) { return a.x < b.x; });
    sort(qs.begin(), qs.end(),
         [](const Q &a, const Q &b) { return a.x < b.x; });
    int j = 0;
    BIT bit(ys.size());
    for (auto &query : qs) {
      while (j < n + n and rs[j].x < query.x) {
        auto &p = rectangles[rs[j].idx];
        if (rs[j].type) {
          bit.apply(rs[j].d, {-p.w * p.r * p.d, -p.w, p.w * p.d, p.w * p.r});
          bit.apply(rs[j].u, {p.w * p.r * p.u, p.w, -p.w * p.u, -p.w * p.r});
        } else {
          bit.apply(rs[j].d, {p.w * p.l * p.d, p.w, -p.w * p.d, -p.w * p.l});
          bit.apply(rs[j].u, {-p.w * p.l * p.u, -p.w, p.w * p.u, p.w * p.l});
        }
        ++j;
      }
      auto &p = queries[query.idx];
      auto uret = bit.prod(query.u);
      ans[query.idx] += uret[0];
      ans[query.idx] += uret[1] * query.x * p.u;
      ans[query.idx] += uret[2] * query.x;
      ans[query.idx] += uret[3] * p.u;
      auto dret = bit.prod(query.d);
      ans[query.idx] -= dret[0];
      ans[query.idx] -= dret[1] * query.x * p.d;
      ans[query.idx] -= dret[2] * query.x;
      ans[query.idx] -= dret[3] * p.d;
      if (not query.type) ans[query.idx] *= -1;
    }
    return ans;
  }
};