Segment Tree (Fractinal Cascading) (structure/segment-tree/segment-tree-fractional-cascading.hpp)

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• Last update: 2024-08-10 03:11:07+09:00
• Include: #include "structure/segment-tree/segment-tree-fractional-cascading.hpp"

二次元平面上の長方形内の点の個数を数えるクエリをセグメント木によって求めます。

Fractional Cascading による高速化により、クエリあたり $O(\log n)$ で処理できます。

コンストラクタ

SegmentTreeFractionalCascading< T >(const vector<T> &v) 

1. 配列 v で初期化します。

計算量

• $O(n \log n)$

prod

S prod(int a, int b, T l, T r) const

$a \leq k \lt b$ かつ $l \leq v_k \lt r$ を満たす $k$ の個数を返します。

制約

• $0 \leq a \leq b \leq n$

計算量

• $O(\log n)$

Verified with

• test/verify/aoj-2674-3.test.cpp

Code

template <typename T>
struct SegmentTreeFractionalCascading {
 private:
  vector<vector<T> > seg;
  vector<vector<int> > LL, RR;
  int sz;

  int prod(int a, int b, int lower, int upper, int k, int l, int r) const {
    if (a >= r or b <= l) {
      return 0;
    } else if (a <= l and r <= b) {
      return upper - lower;
    } else {
      return prod(a, b, LL[k][lower], LL[k][upper], 2 * k + 1, l,
                  (l + r) >> 1) +
             prod(a, b, RR[k][lower], RR[k][upper], 2 * k + 2, (l + r) >> 1, r);
    }
  }

 public:
  SegmentTreeFractionalCascading() = default;

  explicit SegmentTreeFractionalCascading(const vector<T> &v) {
    sz = 1;
    while (sz < v.size()) sz <<= 1;
    seg.resize(2 * sz - 1);
    LL.resize(2 * sz - 1);
    RR.resize(2 * sz - 1);
    for (int k = 0; k < v.size(); k++) {
      seg[k + sz - 1].emplace_back(v[k]);
    }
    for (int k = sz - 2; k >= 0; k--) {
      seg[k].resize(seg[2 * k + 1].size() + seg[2 * k + 2].size());
      LL[k].resize(seg[k].size() + 1);
      RR[k].resize(seg[k].size() + 1);
      merge(seg[2 * k + 1].begin(), seg[2 * k + 1].end(),
            seg[2 * k + 2].begin(), seg[2 * k + 2].end(), seg[k].begin());
      int tail1 = 0, tail2 = 0;
      for (int i = 0; i < seg[k].size(); i++) {
        while (tail1 < seg[2 * k + 1].size() and
               seg[2 * k + 1][tail1] < seg[k][i])
          ++tail1;
        while (tail2 < seg[2 * k + 2].size() and
               seg[2 * k + 2][tail2] < seg[k][i])
          ++tail2;
        LL[k][i] = tail1, RR[k][i] = tail2;
      }
      LL[k][seg[k].size()] = (int)seg[2 * k + 1].size();
      RR[k][seg[k].size()] = (int)seg[2 * k + 2].size();
    }
  }

  int prod(int a, int b, T l, T r) const {
    l = int(lower_bound(seg[0].begin(), seg[0].end(), l) - seg[0].begin());
    r = int(lower_bound(seg[0].begin(), seg[0].end(), r) - seg[0].begin());
    return prod(a, b, l, r, 0, 0, sz);
  }
};

#line 1 "structure/segment-tree/segment-tree-fractional-cascading.hpp"
template <typename T>
struct SegmentTreeFractionalCascading {
 private:
  vector<vector<T> > seg;
  vector<vector<int> > LL, RR;
  int sz;

  int prod(int a, int b, int lower, int upper, int k, int l, int r) const {
    if (a >= r or b <= l) {
      return 0;
    } else if (a <= l and r <= b) {
      return upper - lower;
    } else {
      return prod(a, b, LL[k][lower], LL[k][upper], 2 * k + 1, l,
                  (l + r) >> 1) +
             prod(a, b, RR[k][lower], RR[k][upper], 2 * k + 2, (l + r) >> 1, r);
    }
  }

 public:
  SegmentTreeFractionalCascading() = default;

  explicit SegmentTreeFractionalCascading(const vector<T> &v) {
    sz = 1;
    while (sz < v.size()) sz <<= 1;
    seg.resize(2 * sz - 1);
    LL.resize(2 * sz - 1);
    RR.resize(2 * sz - 1);
    for (int k = 0; k < v.size(); k++) {
      seg[k + sz - 1].emplace_back(v[k]);
    }
    for (int k = sz - 2; k >= 0; k--) {
      seg[k].resize(seg[2 * k + 1].size() + seg[2 * k + 2].size());
      LL[k].resize(seg[k].size() + 1);
      RR[k].resize(seg[k].size() + 1);
      merge(seg[2 * k + 1].begin(), seg[2 * k + 1].end(),
            seg[2 * k + 2].begin(), seg[2 * k + 2].end(), seg[k].begin());
      int tail1 = 0, tail2 = 0;
      for (int i = 0; i < seg[k].size(); i++) {
        while (tail1 < seg[2 * k + 1].size() and
               seg[2 * k + 1][tail1] < seg[k][i])
          ++tail1;
        while (tail2 < seg[2 * k + 2].size() and
               seg[2 * k + 2][tail2] < seg[k][i])
          ++tail2;
        LL[k][i] = tail1, RR[k][i] = tail2;
      }
      LL[k][seg[k].size()] = (int)seg[2 * k + 1].size();
      RR[k][seg[k].size()] = (int)seg[2 * k + 2].size();
    }
  }

  int prod(int a, int b, T l, T r) const {
    l = int(lower_bound(seg[0].begin(), seg[0].end(), l) - seg[0].begin());
    r = int(lower_bound(seg[0].begin(), seg[0].end(), r) - seg[0].begin());
    return prod(a, b, l, r, 0, 0, sz);
  }
};

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