Area of Union of Rectangles (長方形の和集合の面積) (other/area-of-union-of-rectangles.hpp)

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• Last update: 2024-08-10 03:11:07+09:00
• Include: #include "other/area-of-union-of-rectangles.hpp"

いくつかの長方形が与えられたときに、長方形の和集合の面積を求めます。

遅延伝搬セグメント木に、最小値と最小値の個数を求めるモノイドをのせます。

コンストラクタ

(1) AreaOfUnionOfRectangles< T >()
(2) AreaOfUnionOfRectangles< T >(int n)

T は座標が収まる型を指定してください。

(2) で長方形の個数 $n$ 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)

$\lbrace (x,y):l \leq x \leq r, d \leq y \leq u\rbrace$ で表される長方形を追加します。

制約

• $l \lt r$
• $d \lt u$

計算量

• $O(1)$

calc

template<typename T2>
T2 calc() const

長方形の和集合の面積を返します。

T2 は面積が収まる型を指定してください。

計算量

• $O(n \log n)$

Depends on

• structure/class/acted-monoid.hpp
• Lazy Segment Tree (遅延伝搬セグメント木) (structure/segment-tree/lazy-segment-tree.hpp)

Verified with

• test/verify/yosupo-area-of-union-of-rectangles.test.cpp

Code

#include "../structure/segment-tree/lazy-segment-tree.hpp"

template <typename T>
struct AreaOfUnionOfRectangles {
 private:
  struct Rectangle {
    T l, d, r, u;
  };
  vector<Rectangle> rectangles;

 public:
  AreaOfUnionOfRectangles() = default;
  explicit AreaOfUnionOfRectangles(int n) : rectangles{} {
    rectangles.reserve(n);
  }
  void add_rectangle(T l, T d, T r, T u) {
    assert(l < r and d < u);
    rectangles.emplace_back(Rectangle{l, d, r, u});
  }

  template <typename T2>
  T2 calc() const {
    int n = (int)rectangles.size();
    if (n == 0) return 0;

    vector<T> ys;
    vector<tuple<T, int, int>> xs;
    ys.reserve(n + n);
    xs.reserve(n + n);
    for (int i = 0; i < n; i++) {
      auto &rect = rectangles[i];
      ys.emplace_back(rect.d);
      ys.emplace_back(rect.u);
      xs.emplace_back(rect.l, i, +1);
      xs.emplace_back(rect.r, i, -1);
    }
    sort(ys.begin(), ys.end());
    ys.erase(unique(ys.begin(), ys.end()), ys.end());
    sort(xs.begin(), xs.end());
    vector<int> to_d(n), to_u(n);
    for (int i = 0; i < n; i++) {
      auto &rect = rectangles[i];
      to_d[i] = lower_bound(ys.begin(), ys.end(), rect.d) - ys.begin();
      to_u[i] = lower_bound(ys.begin(), ys.end(), rect.u) - ys.begin();
    }
    using pi = pair<int, T>;
    auto f = [](const pi &a, const pi &b) -> pi {
      if (a.first < b.first) return a;
      if (b.first < a.first) return b;
      return {a.first, a.second + b.second};
    };
    auto e = [&]() -> pi { return {n + 1, 0}; };
    auto g = [](const pi &a, int b) -> pi { return {a.first + b, a.second}; };
    auto h = [](int a, int b) -> int { return a + b; };
    auto id = []() { return 0; };
    vector<pi> vs(ys.size() - 1);
    for (int i = 0; i + 1 < ys.size(); i++) {
      vs[i] = {0, ys[i + 1] - ys[i]};
    }
    LazySegmentTree seg(LambdaActedMonoid(f, e, g, h, id), vs);
    T2 ret = 0;
    T total = ys.back() - ys.front();
    for (int i = 0; i + 1 < n + n; i++) {
      auto &[k, j, d] = xs[i];
      seg.apply(to_d[j], to_u[j], d);
      auto [v, cnt] = seg.all_prod();
      T2 dy = total - (v == 0 ? cnt : 0);
      T2 dx = get<0>(xs[i + 1]) - k;
      ret += dy * dx;
    }
    return ret;
  }
};

#line 2 "structure/class/acted-monoid.hpp"

template <typename S2, typename Op, typename E, typename F2, typename Mapping,
          typename Composition, typename Id>
struct LambdaActedMonoid {
  using S = S2;
  using F = F2;

  S op(const S &a, const S &b) const { return _op(a, b); }

  S e() const { return _e(); }

  S mapping(const S &x, const F &f) const { return _mapping(x, f); }

  F composition(const F &f, const F &g) const { return _composition(f, g); }

  F id() const { return _id(); }

  LambdaActedMonoid(Op _op, E _e, Mapping _mapping, Composition _composition,
                    Id _id)
      : _op(_op),
        _e(_e),
        _mapping(_mapping),
        _composition(_composition),
        _id(_id) {}

 private:
  Op _op;

  E _e;

  Mapping _mapping;

  Composition _composition;

  Id _id;
};

template <typename Op, typename E, typename Mapping, typename Composition,
          typename Id>
LambdaActedMonoid(Op _op, E _e, Mapping _mapping, Composition _composition,
                  Id _id)
    -> LambdaActedMonoid<decltype(_e()), Op, E, decltype(_id()), Mapping,
                         Composition, Id>;

/*
struct ActedMonoid {
  using S = ?;
  using F = ?;
  static constexpr S op(const S& a, const S& b) {}
  static constexpr S e() {}
  static constexpr S mapping(const S &x, const F &f) {}
  static constexpr F composition(const F &f, const F &g) {}
  static constexpr F id() {}
};
*/
#line 2 "structure/segment-tree/lazy-segment-tree.hpp"

template <typename ActedMonoid>
struct LazySegmentTree {
  using S = typename ActedMonoid::S;
  using F = typename ActedMonoid::F;

 private:
  ActedMonoid m;

  int n{}, sz{}, height{};

  vector<S> data;

  vector<F> lazy;

  inline void update(int k) {
    data[k] = m.op(data[2 * k + 0], data[2 * k + 1]);
  }

  inline void all_apply(int k, const F &x) {
    data[k] = m.mapping(data[k], x);
    if (k < sz) lazy[k] = m.composition(lazy[k], x);
  }

  inline void propagate(int k) {
    if (lazy[k] != m.id()) {
      all_apply(2 * k + 0, lazy[k]);
      all_apply(2 * k + 1, lazy[k]);
      lazy[k] = m.id();
    }
  }

 public:
  LazySegmentTree() = default;

  explicit LazySegmentTree(ActedMonoid m, int n) : m(m), n(n) {
    sz = 1;
    height = 0;
    while (sz < n) sz <<= 1, height++;
    data.assign(2 * sz, m.e());
    lazy.assign(2 * sz, m.id());
  }

  explicit LazySegmentTree(ActedMonoid m, const vector<S> &v)
      : LazySegmentTree(m, v.size()) {
    build(v);
  }

  void build(const vector<S> &v) {
    assert(n == (int)v.size());
    for (int k = 0; k < n; k++) data[k + sz] = v[k];
    for (int k = sz - 1; k > 0; k--) update(k);
  }

  void set(int k, const S &x) {
    k += sz;
    for (int i = height; i > 0; i--) propagate(k >> i);
    data[k] = x;
    for (int i = 1; i <= height; i++) update(k >> i);
  }

  S get(int k) {
    k += sz;
    for (int i = height; i > 0; i--) propagate(k >> i);
    return data[k];
  }

  S operator[](int k) { return get(k); }

  S prod(int l, int r) {
    if (l >= r) return m.e();
    l += sz;
    r += sz;
    for (int i = height; i > 0; i--) {
      if (((l >> i) << i) != l) propagate(l >> i);
      if (((r >> i) << i) != r) propagate((r - 1) >> i);
    }
    S L = m.e(), R = m.e();
    for (; l < r; l >>= 1, r >>= 1) {
      if (l & 1) L = m.op(L, data[l++]);
      if (r & 1) R = m.op(data[--r], R);
    }
    return m.op(L, R);
  }

  S all_prod() const { return data[1]; }

  void apply(int k, const F &f) {
    k += sz;
    for (int i = height; i > 0; i--) propagate(k >> i);
    data[k] = m.mapping(data[k], f);
    for (int i = 1; i <= height; i++) update(k >> i);
  }

  void apply(int l, int r, const F &f) {
    if (l >= r) return;
    l += sz;
    r += sz;
    for (int i = height; i > 0; i--) {
      if (((l >> i) << i) != l) propagate(l >> i);
      if (((r >> i) << i) != r) propagate((r - 1) >> i);
    }
    {
      int l2 = l, r2 = r;
      for (; l < r; l >>= 1, r >>= 1) {
        if (l & 1) all_apply(l++, f);
        if (r & 1) all_apply(--r, f);
      }
      l = l2, r = r2;
    }
    for (int i = 1; i <= height; i++) {
      if (((l >> i) << i) != l) update(l >> i);
      if (((r >> i) << i) != r) update((r - 1) >> i);
    }
  }

  template <typename C>
  int find_first(int l, const C &check) {
    if (l >= n) return n;
    l += sz;
    for (int i = height; i > 0; i--) propagate(l >> i);
    S sum = m.e();
    do {
      while ((l & 1) == 0) l >>= 1;
      if (check(m.op(sum, data[l]))) {
        while (l < sz) {
          propagate(l);
          l <<= 1;
          auto nxt = m.op(sum, data[l]);
          if (not check(nxt)) {
            sum = nxt;
            l++;
          }
        }
        return l + 1 - sz;
      }
      sum = m.op(sum, data[l++]);
    } while ((l & -l) != l);
    return n;
  }

  template <typename C>
  int find_last(int r, const C &check) {
    if (r <= 0) return -1;
    r += sz;
    for (int i = height; i > 0; i--) propagate((r - 1) >> i);
    S sum = m.e();
    do {
      r--;
      while (r > 1 and (r & 1)) r >>= 1;
      if (check(m.op(data[r], sum))) {
        while (r < sz) {
          propagate(r);
          r = (r << 1) + 1;
          auto nxt = m.op(data[r], sum);
          if (not check(nxt)) {
            sum = nxt;
            r--;
          }
        }
        return r - sz;
      }
      sum = m.op(data[r], sum);
    } while ((r & -r) != r);
    return -1;
  }
};
#line 2 "other/area-of-union-of-rectangles.hpp"

template <typename T>
struct AreaOfUnionOfRectangles {
 private:
  struct Rectangle {
    T l, d, r, u;
  };
  vector<Rectangle> rectangles;

 public:
  AreaOfUnionOfRectangles() = default;
  explicit AreaOfUnionOfRectangles(int n) : rectangles{} {
    rectangles.reserve(n);
  }
  void add_rectangle(T l, T d, T r, T u) {
    assert(l < r and d < u);
    rectangles.emplace_back(Rectangle{l, d, r, u});
  }

  template <typename T2>
  T2 calc() const {
    int n = (int)rectangles.size();
    if (n == 0) return 0;

    vector<T> ys;
    vector<tuple<T, int, int>> xs;
    ys.reserve(n + n);
    xs.reserve(n + n);
    for (int i = 0; i < n; i++) {
      auto &rect = rectangles[i];
      ys.emplace_back(rect.d);
      ys.emplace_back(rect.u);
      xs.emplace_back(rect.l, i, +1);
      xs.emplace_back(rect.r, i, -1);
    }
    sort(ys.begin(), ys.end());
    ys.erase(unique(ys.begin(), ys.end()), ys.end());
    sort(xs.begin(), xs.end());
    vector<int> to_d(n), to_u(n);
    for (int i = 0; i < n; i++) {
      auto &rect = rectangles[i];
      to_d[i] = lower_bound(ys.begin(), ys.end(), rect.d) - ys.begin();
      to_u[i] = lower_bound(ys.begin(), ys.end(), rect.u) - ys.begin();
    }
    using pi = pair<int, T>;
    auto f = [](const pi &a, const pi &b) -> pi {
      if (a.first < b.first) return a;
      if (b.first < a.first) return b;
      return {a.first, a.second + b.second};
    };
    auto e = [&]() -> pi { return {n + 1, 0}; };
    auto g = [](const pi &a, int b) -> pi { return {a.first + b, a.second}; };
    auto h = [](int a, int b) -> int { return a + b; };
    auto id = []() { return 0; };
    vector<pi> vs(ys.size() - 1);
    for (int i = 0; i + 1 < ys.size(); i++) {
      vs[i] = {0, ys[i + 1] - ys[i]};
    }
    LazySegmentTree seg(LambdaActedMonoid(f, e, g, h, id), vs);
    T2 ret = 0;
    T total = ys.back() - ys.front();
    for (int i = 0; i + 1 < n + n; i++) {
      auto &[k, j, d] = xs[i];
      seg.apply(to_d[j], to_u[j], d);
      auto [v, cnt] = seg.all_prod();
      T2 dy = total - (v == 0 ? cnt : 0);
      T2 dx = get<0>(xs[i + 1]) - k;
      ret += dy * dx;
    }
    return ret;
  }
};

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