Area of Union of Rectangles
(data-structure/others/AreaofUnionofRectangles.hpp)

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• Last update: 2024-11-22 13:46:17+09:00
• Include: #include "data-structure/others/AreaofUnionofRectangles.hpp"

概要

todo

計算量

todo

Depends on

• Lazy Segment Tree (data-structure/segment-tree/LazySegmentTree.hpp)

Verified with

• verify/LibraryChecker/data-structure/others/AreaofUnionofRectangles.test.cpp

Code

#pragma once
#include "./../segment-tree/LazySegmentTree.hpp"
using Sa = pair<int, long long>;
using Fa = int;
Sa opa(Sa a, Sa b) {
  if (a.first < b.first) return a;
  if (a.first > b.first) return b;
  return {a.first, a.second + b.second};
}

Sa ea() {
  return {1e9 + 1e5, 0};
}

Sa mappinga(Fa f, Sa x) {
  return {x.first + f, x.second};
}

Fa compositiona(Fa f, Fa g) {
  return f + g;
}

Fa ida() {
  return 0;
}

class AreaofUnionofRectangles {
 private:
  vector<long long> x1, x2, y1, y2;
  int n;
  lint res = 0;

 public:
  AreaofUnionofRectangles(const vector<long long> &xl, const vector<long long> &xr, const vector<long long> &yl, const vector<long long> &yr) {
    x1 = xl;
    x2 = xr;
    y1 = yl;
    y2 = yr;
    n = x1.size();
    vector<int> cmp(n * 2);
    vector<tuple<long long, int, int, int>> q(n * 2);
    for (int i = 0; i < n; i++) {
      cmp[i * 2] = y1[i];
      cmp[i * 2 + 1] = y2[i];
    }
    sort(cmp.begin(), cmp.end());
    cmp.erase(unique(cmp.begin(), cmp.end()), cmp.end());
    for (int i = 0; i < n; i++) {
      int idx1 = distance(cmp.begin(), lower_bound(cmp.begin(), cmp.end(), y1[i]));
      int idx2 = distance(cmp.begin(), lower_bound(cmp.begin(), cmp.end(), y2[i]));
      q[i * 2] = {x1[i], 1, idx1, idx2};
      q[i * 2 + 1] = {x2[i], -1, idx1, idx2};
    }

    sort(q.begin(), q.end());
    int siz = cmp.size() - 1;
    vector<Sa> v(siz);
    for (int i = 0; i < siz; i++) {
      v[i] = {0, cmp[i + 1] - cmp[i]};
    }

    lazy_segtree<Sa, opa, ea, Fa, mappinga, compositiona, ida> seg(v);
    long long prev = get<0>(q[0]);
    for (int i = 0; i < n * 2; i++) {
      long long x = get<0>(q[i]);
      int f = get<1>(q[i]);
      long long l = get<2>(q[i]);
      long long r = get<3>(q[i]);
      long long h = x - prev;
      long long w = cmp[siz] - cmp[0];
      Sa pr = seg.all_prod();
      if (pr.first == 0) w -= pr.second;
      res += h * w;
      prev = x;
      seg.apply(l, r, f);
    }
  }

  lint ans() {
    return res;
  }
};

#line 1 "data-structure/segment-tree/LazySegmentTree.hpp"

template <class S,
          auto op,
          auto e,
          class F,
          auto mapping,
          auto composition,
          auto id>
struct lazy_segtree {
 private:
  unsigned int seg_bit_ceil(unsigned int n) {
    unsigned int x = 1;
    while (x < (unsigned int)(n)) x *= 2;
    return x;
  }

 public:
  static_assert(std::is_convertible_v<decltype(op), std::function<S(S, S)>>,
                "op must work as S(S, S)");
  static_assert(std::is_convertible_v<decltype(e), std::function<S()>>,
                "e must work as S()");
  static_assert(
      std::is_convertible_v<decltype(mapping), std::function<S(F, S)>>,
      "mapping must work as F(F, S)");
  static_assert(
      std::is_convertible_v<decltype(composition), std::function<F(F, F)>>,
      "compostiion must work as F(F, F)");
  static_assert(std::is_convertible_v<decltype(id), std::function<F()>>,
                "id must work as F()");
  lazy_segtree() : lazy_segtree(0) {}
  explicit lazy_segtree(int n) : lazy_segtree(std::vector<S>(n, e())) {}
  explicit lazy_segtree(const std::vector<S>& v) : _n(int(v.size())) {
    size = (int)seg_bit_ceil((unsigned int)(_n));
    log = __builtin_ctz((unsigned int)size);
    d = std::vector<S>(2 * size, e());
    lz = std::vector<F>(size, id());
    for (int i = 0; i < _n; i++) d[size + i] = v[i];
    for (int i = size - 1; i >= 1; i--) {
      update(i);
    }
  }

  void set(int p, S x) {
    assert(0 <= p && p < _n);
    p += size;
    for (int i = log; i >= 1; i--) push(p >> i);
    d[p] = x;
    for (int i = 1; i <= log; i++) update(p >> i);
  }

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

  S prod(int l, int r) {
    assert(0 <= l && l <= r && r <= _n);
    if (l == r) return e();

    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);
    }

    S sml = e(), smr = e();
    while (l < r) {
      if (l & 1) sml = op(sml, d[l++]);
      if (r & 1) smr = op(d[--r], smr);
      l >>= 1;
      r >>= 1;
    }

    return op(sml, smr);
  }

  S all_prod() { return d[1]; }

  void apply(int p, F f) {
    assert(0 <= p && p < _n);
    p += size;
    for (int i = log; i >= 1; i--) push(p >> i);
    d[p] = mapping(f, d[p]);
    for (int i = 1; i <= log; i++) update(p >> i);
  }
  void apply(int l, int r, F f) {
    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) all_apply(l++, f);
        if (r & 1) all_apply(--r, f);
        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 <bool (*g)(S)>
  int max_right(int l) {
    return max_right(l, [](S x) { return g(x); });
  }
  template <class G>
  int max_right(int l, G g) {
    assert(0 <= l && l <= _n);
    assert(g(e()));
    if (l == _n) return _n;
    l += size;
    for (int i = log; i >= 1; i--) push(l >> i);
    S sm = e();
    do {
      while (l % 2 == 0) l >>= 1;
      if (!g(op(sm, d[l]))) {
        while (l < size) {
          push(l);
          l = (2 * l);
          if (g(op(sm, d[l]))) {
            sm = op(sm, d[l]);
            l++;
          }
        }
        return l - size;
      }
      sm = op(sm, d[l]);
      l++;
    } while ((l & -l) != l);
    return _n;
  }

  template <bool (*g)(S)>
  int min_left(int r) {
    return min_left(r, [](S x) { return g(x); });
  }
  template <class G>
  int min_left(int r, G g) {
    assert(0 <= r && r <= _n);
    assert(g(e()));
    if (r == 0) return 0;
    r += size;
    for (int i = log; i >= 1; i--) push((r - 1) >> i);
    S sm = e();
    do {
      r--;
      while (r > 1 && (r % 2)) r >>= 1;
      if (!g(op(d[r], sm))) {
        while (r < size) {
          push(r);
          r = (2 * r + 1);
          if (g(op(d[r], sm))) {
            sm = op(d[r], sm);
            r--;
          }
        }
        return r + 1 - size;
      }
      sm = op(d[r], sm);
    } while ((r & -r) != r);
    return 0;
  }

 private:
  int _n, size, log;
  std::vector<S> d;
  std::vector<F> lz;

  void update(int k) { d[k] = op(d[2 * k], d[2 * k + 1]); }
  void all_apply(int k, F f) {
    d[k] = mapping(f, d[k]);
    if (k < size) lz[k] = composition(f, lz[k]);
  }
  void push(int k) {
    all_apply(2 * k, lz[k]);
    all_apply(2 * k + 1, lz[k]);
    lz[k] = id();
  }
};
#line 3 "data-structure/others/AreaofUnionofRectangles.hpp"
using Sa = pair<int, long long>;
using Fa = int;
Sa opa(Sa a, Sa b) {
  if (a.first < b.first) return a;
  if (a.first > b.first) return b;
  return {a.first, a.second + b.second};
}

Sa ea() {
  return {1e9 + 1e5, 0};
}

Sa mappinga(Fa f, Sa x) {
  return {x.first + f, x.second};
}

Fa compositiona(Fa f, Fa g) {
  return f + g;
}

Fa ida() {
  return 0;
}

class AreaofUnionofRectangles {
 private:
  vector<long long> x1, x2, y1, y2;
  int n;
  lint res = 0;

 public:
  AreaofUnionofRectangles(const vector<long long> &xl, const vector<long long> &xr, const vector<long long> &yl, const vector<long long> &yr) {
    x1 = xl;
    x2 = xr;
    y1 = yl;
    y2 = yr;
    n = x1.size();
    vector<int> cmp(n * 2);
    vector<tuple<long long, int, int, int>> q(n * 2);
    for (int i = 0; i < n; i++) {
      cmp[i * 2] = y1[i];
      cmp[i * 2 + 1] = y2[i];
    }
    sort(cmp.begin(), cmp.end());
    cmp.erase(unique(cmp.begin(), cmp.end()), cmp.end());
    for (int i = 0; i < n; i++) {
      int idx1 = distance(cmp.begin(), lower_bound(cmp.begin(), cmp.end(), y1[i]));
      int idx2 = distance(cmp.begin(), lower_bound(cmp.begin(), cmp.end(), y2[i]));
      q[i * 2] = {x1[i], 1, idx1, idx2};
      q[i * 2 + 1] = {x2[i], -1, idx1, idx2};
    }

    sort(q.begin(), q.end());
    int siz = cmp.size() - 1;
    vector<Sa> v(siz);
    for (int i = 0; i < siz; i++) {
      v[i] = {0, cmp[i + 1] - cmp[i]};
    }

    lazy_segtree<Sa, opa, ea, Fa, mappinga, compositiona, ida> seg(v);
    long long prev = get<0>(q[0]);
    for (int i = 0; i < n * 2; i++) {
      long long x = get<0>(q[i]);
      int f = get<1>(q[i]);
      long long l = get<2>(q[i]);
      long long r = get<3>(q[i]);
      long long h = x - prev;
      long long w = cmp[siz] - cmp[0];
      Sa pr = seg.all_prod();
      if (pr.first == 0) w -= pr.second;
      res += h * w;
      prev = x;
      seg.apply(l, r, f);
    }
  }

  lint ans() {
    return res;
  }
};

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