ds/rectangle_union.hpp

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• Last update: 2026-09-17 11:49:38+09:00
• Include: #include "ds/rectangle_union.hpp"

Depends on

• alg/acted_monoid/minmincnt_add.hpp
• alg/monoid/add.hpp
• alg/monoid/minmincnt.hpp
• ds/segtree/lazy_segtree.hpp
• other/bit.hpp

Verified with

• test/2_library_checker/data_structure/area_of_union_of_rectangles.test.cpp
• test/4_aoj/DSL_4_A.test.cpp

Code

#include "ds/segtree/lazy_segtree.hpp"

#include "alg/acted_monoid/minmincnt_add.hpp"


template <typename XY = int>
struct Rectangle_Union {
  using RECT = tuple<XY, XY, XY, XY>;
  vc<RECT> rectangles;
  vc<XY> X, Y;

  void add_rect(XY xl, XY xr, XY yl, XY yr) {
    assert(xl < xr && yl < yr);
    X.eb(xl), X.eb(xr), Y.eb(yl), Y.eb(yr);
    rectangles.eb(xl, xr, yl, yr);
  }

  template <typename ANS_TYPE = ll>
  ANS_TYPE calc() {
    if (rectangles.empty()) return 0;
    int N = len(X);
    vc<int> ord_x = argsort(X);
    vc<int> ord_y = argsort(Y);
    vc<int> rk_y(N);
    FOR(i, N) rk_y[ord_y[i]] = i;
    X = rearrange(X, ord_x);
    Y = rearrange(Y, ord_y);

    using AM = ActedMonoid_MinMincnt_Add<XY>;
    Lazy_SegTree<AM> seg(N - 1, [&](int i) -> pair<XY, XY> { return {0, Y[i + 1] - Y[i]}; });

    ANS_TYPE ANS = 0;
    XY total = Y.back() - Y[0];
    FOR(i, N - 1) {
      int k = ord_x[i] / 2;
      int a = (ord_x[i] & 1 ? -1 : 1);
      seg.apply(rk_y[2 * k], rk_y[2 * k + 1], a);
      auto [min, mincnt] = seg.prod_all();
      ANS_TYPE dy = total - (min == 0 ? mincnt : 0);
      ANS_TYPE dx = X[i + 1] - X[i];
      ANS += dx * dy;
    }
    return ANS;
  }
};

#line 1 "other/bit.hpp"

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_sgn(int x) { return (__builtin_parity(unsigned(x)) & 1 ? -1 : 1); }
int popcnt_sgn(u32 x) { return (__builtin_parity(x) & 1 ? -1 : 1); }
int popcnt_sgn(ll x) { return (__builtin_parityll(x) & 1 ? -1 : 1); }
int popcnt_sgn(u64 x) { return (__builtin_parityll(x) & 1 ? -1 : 1); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T>
T kth_bit(int k) {
  assert(0 <= k && k < int(8 * sizeof(T)));
  return T(1) << k;
}
template <typename T>
bool has_kth_bit(T x, int k) {
  assert(0 <= k && k < int(8 * sizeof(T)));
  return x >> k & 1;
}

template <typename UINT>
struct all_bit {
  static_assert(is_unsigned<UINT>::value);
  UINT s;
  all_bit(UINT s) : s(s) {}
  struct iter {
    UINT s;
    int operator*() const { return lowbit(s); }
    void operator++() { s &= s - 1; }
    bool operator!=(nullptr_t) const { return s; }
  };
  iter begin() const { return {s}; }
  nullptr_t end() const { return nullptr; }
};

template <typename UINT>
struct all_subset {
  static_assert(is_unsigned<UINT>::value);
  UINT s;
  all_subset(UINT s) : s(s) {}
  struct iter {
    UINT s, t;
    bool done = false;
    UINT operator*() const { return t; }
    void operator++() {
      done = (t == 0);
      t = (t - 1) & s;
    }
    bool operator!=(nullptr_t) const { return !done; }
  };
  iter begin() const { return {s, s}; }
  nullptr_t end() const { return nullptr; }
};

constexpr u64 full_mask(int n) {
  assert(0 <= n && n <= 64);
  return n == 64 ? -1ULL : (1ULL << n) - 1;
}

u64 bit_reverse(u64 x) {
  x = ((x & 0x5555555555555555ULL) << 1) | ((x >> 1) & 0x5555555555555555ULL);
  x = ((x & 0x3333333333333333ULL) << 2) | ((x >> 2) & 0x3333333333333333ULL);
  x = ((x & 0x0f0f0f0f0f0f0f0fULL) << 4) | ((x >> 4) & 0x0f0f0f0f0f0f0f0fULL);
  x = ((x & 0x00ff00ff00ff00ffULL) << 8) | ((x >> 8) & 0x00ff00ff00ff00ffULL);
  x = ((x & 0x0000ffff0000ffffULL) << 16) | ((x >> 16) & 0x0000ffff0000ffffULL);
  x = (x << 32) | (x >> 32);
  return x;
}
#line 2 "ds/segtree/lazy_segtree.hpp"

template <typename ActedMonoid>
struct Lazy_SegTree {
  using AM = ActedMonoid;
  using MX = typename AM::Monoid_X;
  using MA = typename AM::Monoid_A;
  using X = typename MX::value_type;
  using A = typename MA::value_type;
  int n, log, size;
  vc<X> dat;
  vc<A> laz;
  vc<bool> has_laz;

  Lazy_SegTree() {}
  Lazy_SegTree(int n) { build(n); }
  template <typename F>
  Lazy_SegTree(int n, F f) {
    build(n, f);
  }
  Lazy_SegTree(const vc<X>& v) { build(v); }

  void build(int m) {
    build(m, [](int i) -> X { return MX::id(); });
  }
  void build(const vc<X>& v) {
    build(len(v), [&](int i) -> X { return v[i]; });
  }
  template <typename F>
  void build(int m, F f) {
    n = m, log = 0;
    while ((1 << log) < n) ++log;
    size = 1 << log;
    dat.assign(size << 1, MX::id());
    laz.assign(size, MA::id());
    has_laz.assign(size, false);
    FOR(i, n) dat[size + i] = f(i);
    FOR_R(i, 1, size) update(i);
  }

  void update(int k) { dat[k] = MX::op(dat[2 * k], dat[2 * k + 1]); }
  void set(int p, X x) {
    assert(0 <= p && p < n);
    p += size;
    for (int i = log; i >= 1; i--) push(p >> i);
    dat[p] = x;
    for (int i = 1; i <= log; i++) update(p >> i);
  }
  void multiply(int p, const X& x) {
    assert(0 <= p && p < n);
    p += size;
    for (int i = log; i >= 1; i--) push(p >> i);
    dat[p] = MX::op(dat[p], x);
    for (int i = 1; i <= log; i++) update(p >> i);
  }

  X get(int p) {
    assert(0 <= p && p < n);
    p += size;
    for (int i = log; i >= 1; i--) push(p >> i);
    return dat[p];
  }

  vc<X> get_all() {
    FOR(k, 1, size) { push(k); }
    return {dat.begin() + size, dat.begin() + size + n};
  }

  X prod(int l, int r) {
    assert(0 <= l && l <= r && r <= n);
    if (l == r) return MX::id();
    l += size, r += size;
    for (int i = log; i >= 1; i--) {
      if (((l >> i) << i) != l) push(l >> i);
      if (((r >> i) << i) != r) push((r - 1) >> i);
    }
    X xl = MX::id(), xr = MX::id();
    while (l < r) {
      if (l & 1) xl = MX::op(xl, dat[l++]);
      if (r & 1) xr = MX::op(dat[--r], xr);
      l >>= 1, r >>= 1;
    }
    return MX::op(xl, xr);
  }

  X prod_all() { return dat[1]; }

  void apply(int l, int r, A a) {
    assert(0 <= l && l <= r && r <= n);
    if (l == r) return;
    l += size, r += size;
    for (int i = log; i >= 1; i--) {
      if (((l >> i) << i) != l) push(l >> i);
      if (((r >> i) << i) != r) push((r - 1) >> i);
    }
    int l2 = l, r2 = r;
    while (l < r) {
      if (l & 1) apply_at(l++, a);
      if (r & 1) apply_at(--r, a);
      l >>= 1, r >>= 1;
    }
    l = l2, r = r2;
    for (int i = 1; i <= log; i++) {
      if (((l >> i) << i) != l) update(l >> i);
      if (((r >> i) << i) != r) update((r - 1) >> i);
    }
  }

  template <typename F>
  int max_right(const F check, int l) {
    assert(0 <= l && l <= n);
    assert(check(MX::id()));
    if (l == n) return n;
    l += size;
    for (int i = log; i >= 1; i--) push(l >> i);
    X sm = MX::id();
    do {
      while (l % 2 == 0) l >>= 1;
      if (!check(MX::op(sm, dat[l]))) {
        while (l < size) {
          push(l);
          l = (2 * l);
          if (check(MX::op(sm, dat[l]))) {
            sm = MX::op(sm, dat[l++]);
          }
        }
        return l - size;
      }
      sm = MX::op(sm, dat[l++]);
    } while ((l & -l) != l);
    return n;
  }

  template <typename F>
  int min_left(const F check, int r) {
    assert(0 <= r && r <= n);
    assert(check(MX::id()));
    if (r == 0) return 0;
    r += size;
    for (int i = log; i >= 1; i--) push((r - 1) >> i);
    X sm = MX::id();
    do {
      r--;
      while (r > 1 && (r % 2)) r >>= 1;
      if (!check(MX::op(dat[r], sm))) {
        while (r < size) {
          push(r);
          r = (2 * r + 1);
          if (check(MX::op(dat[r], sm))) {
            sm = MX::op(dat[r--], sm);
          }
        }
        return r + 1 - size;
      }
      sm = MX::op(dat[r], sm);
    } while ((r & -r) != r);
    return 0;
  }

  // l <= i xor (xor_val) < r となる i 全体に apply
  void apply_xor_range(int l, int r, int xor_val, A a) {
    assert(!(n & (n - 1)));
    assert(0 <= xor_val && xor_val < n);
    assert(0 <= l && l <= r && r <= n);

    auto dfs = [&](auto& dfs, int idx, int seg_l, int seg_r) -> void {
      if (l <= seg_l && seg_r <= r) {
        return apply_at(idx, a);
      }
      if (r <= seg_l || seg_r <= l) return;
      push(idx);
      int seg_m = (seg_l + seg_r) / 2;
      int bit = (seg_r - seg_l) / 2;
      int left = 2 * idx + 0, right = 2 * idx + 1;
      if (xor_val & bit) swap(left, right);
      dfs(dfs, left, seg_l, seg_m);
      dfs(dfs, right, seg_m, seg_r);
      update(idx);
    };
    dfs(dfs, 1, 0, n);
  }

 private:
  void apply_at(int k, A a) {
    ll sz = 1 << (log - topbit(k));
    dat[k] = AM::act(dat[k], a, sz);
    if (k < size) has_laz[k] = 1, laz[k] = MA::op(laz[k], a);
  }
  void push(int k) {
    if (!has_laz[k]) return;
    has_laz[k] = 0;
    apply_at(2 * k, laz[k]), apply_at(2 * k + 1, laz[k]);
    laz[k] = MA::id();
  }
};
#line 1 "alg/monoid/minmincnt.hpp"

// 最小値、最小値の個数
template <typename E>
struct Monoid_MinMincnt {
  using value_type = pair<E, E>;
  using X = value_type;
  static X op(X x, X y) {
    auto [xmin, xmincnt] = x;
    auto [ymin, ymincnt] = y;
    if (xmin > ymin) return y;
    if (xmin < ymin) return x;
    return {xmin, xmincnt + ymincnt};
  }
  static constexpr X id() { return {infty<E>, 0}; }
  static constexpr bool commute = true;
};
#line 1 "alg/monoid/add.hpp"

template <typename E>
struct Monoid_Add {
  using X = E;
  using value_type = X;
  static constexpr X op(const X &x, const X &y) noexcept { return x + y; }
  static constexpr X inverse(const X &x) noexcept { return -x; }
  static constexpr X power(const X &x, ll n) noexcept { return X(n) * x; }
  static constexpr X id() { return X(0); }
  static constexpr bool commute = true;
};
#line 3 "alg/acted_monoid/minmincnt_add.hpp"

template <typename E>
struct ActedMonoid_MinMincnt_Add {
  using Monoid_X = Monoid_MinMincnt<E>;
  using Monoid_A = Monoid_Add<E>;
  using X = typename Monoid_X::value_type;
  using A = typename Monoid_A::value_type;
  static constexpr X act(const X &x, const A &a, const ll &size) {
    auto [xmin, xmincnt] = x;
    if (xmin == infty<E>) return x;
    return {xmin + a, xmincnt};
  }
};
#line 3 "ds/rectangle_union.hpp"

template <typename XY = int>
struct Rectangle_Union {
  using RECT = tuple<XY, XY, XY, XY>;
  vc<RECT> rectangles;
  vc<XY> X, Y;

  void add_rect(XY xl, XY xr, XY yl, XY yr) {
    assert(xl < xr && yl < yr);
    X.eb(xl), X.eb(xr), Y.eb(yl), Y.eb(yr);
    rectangles.eb(xl, xr, yl, yr);
  }

  template <typename ANS_TYPE = ll>
  ANS_TYPE calc() {
    if (rectangles.empty()) return 0;
    int N = len(X);
    vc<int> ord_x = argsort(X);
    vc<int> ord_y = argsort(Y);
    vc<int> rk_y(N);
    FOR(i, N) rk_y[ord_y[i]] = i;
    X = rearrange(X, ord_x);
    Y = rearrange(Y, ord_y);

    using AM = ActedMonoid_MinMincnt_Add<XY>;
    Lazy_SegTree<AM> seg(N - 1, [&](int i) -> pair<XY, XY> { return {0, Y[i + 1] - Y[i]}; });

    ANS_TYPE ANS = 0;
    XY total = Y.back() - Y[0];
    FOR(i, N - 1) {
      int k = ord_x[i] / 2;
      int a = (ord_x[i] & 1 ? -1 : 1);
      seg.apply(rk_y[2 * k], rk_y[2 * k + 1], a);
      auto [min, mincnt] = seg.prod_all();
      ANS_TYPE dy = total - (min == 0 ? mincnt : 0);
      ANS_TYPE dx = X[i + 1] - X[i];
      ANS += dx * dy;
    }
    return ANS;
  }
};

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