Wavelet Matrix Rectangle Sum (structure/wavelet/wavelet-matrix-rectangle-sum.hpp)

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Last update: 2026-08-10 15:44:16+00:00
Include: #include "structure/wavelet/wavelet-matrix-rectangle-sum.hpp"

静的配列に対して、値範囲での重み総和を処理するデータ構造です。

WaveletMatrixRectangleSum< T, MAXLOG, D > は値配列 v と重み配列 d を受け取り、区間 [l, r) かつ値域 [lower, upper) の重み総和を返します。

コンストラクタ

(1) WaveletMatrixRectangleSum()
(2) WaveletMatrixRectangleSum(const vector<T>& v, const vector<D>& d)

(2) は v[i] に重み d[i] を対応づけて構築します。

制約

v.size() == d.size()

計算量

(2) $O(|v| \cdot MAXLOG)$

rect_sum

(1) D rect_sum(int l, int r, T upper)
(2) D rect_sum(int l, int r, T lower, T upper)

(1) 区間 [l, r) で v[i] < upper を満たす要素の重み総和を返します。
(2) 区間 [l, r) で lower <= v[i] < upper を満たす要素の重み総和を返します。

制約

$0 \leq l \leq r \leq N$

計算量

$O(MAXLOG)$

CompressedWaveletMatrixRectangleSum

座標圧縮を内部で行うラッパです。値が疎な場合はこちらを使うと扱いやすくなります。

コンストラクタ

CompressedWaveletMatrixRectangleSum(const vector<T>& v, const vector<D>& d)

配列 v を座標圧縮して構築します。

制約

v.size() == d.size()

計算量

$O(|v| \log |v| + |v| \cdot MAXLOG)$

rect_sum

(1) D rect_sum(int l, int r, T upper)
(2) D rect_sum(int l, int r, T lower, T upper)

(1) 区間 [l, r) で v[i] < upper を満たす要素の重み総和を返します。
(2) 区間 [l, r) で lower <= v[i] < upper を満たす要素の重み総和を返します。

制約

$0 \leq l \leq r \leq N$

計算量

$O(\log |v| + MAXLOG)$

Depends on

Succinct Indexable Dictionary (完備辞書) (structure/wavelet/succinct-indexable-dictionary.hpp)

Verified with

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

Code

#pragma once

#include <algorithm>
#include <cassert>
#include <cstddef>
#include <iterator>
#include <numeric>
#include <tuple>
#include <utility>
#include <vector>

#include "succinct-indexable-dictionary.hpp"

/*
 * @brief Wavelet Matrix Rectangle Sum
 *
 */
template <typename T, int MAXLOG, typename D>
struct WaveletMatrixRectangleSum {
  std::size_t length;
  SuccinctIndexableDictionary matrix[MAXLOG];
  std::vector<D> ds[MAXLOG];
  int mid[MAXLOG];

  WaveletMatrixRectangleSum() = default;

  WaveletMatrixRectangleSum(const std::vector<T>& v, const std::vector<D>& d)
      : length(v.size()) {
    assert(v.size() == d.size());
    std::vector<int> l(length), r(length), ord(length);
    std::iota(std::begin(ord), std::end(ord), 0);
    for (int level = MAXLOG - 1; level >= 0; level--) {
      matrix[level] = SuccinctIndexableDictionary(length + 1);
      int left = 0, right = 0;
      for (int i = 0; i < length; i++) {
        if (((v[ord[i]] >> level) & 1)) {
          matrix[level].set(i);
          r[right++] = ord[i];
        } else {
          l[left++] = ord[i];
        }
      }
      mid[level] = left;
      matrix[level].build();
      std::swap(ord, l);
      for (int i = 0; i < right; i++) {
        ord[left + i] = r[i];
      }
      ds[level].resize(length + 1);
      ds[level][0] = D();
      for (int i = 0; i < length; i++) {
        ds[level][i + 1] = ds[level][i] + d[ord[i]];
      }
    }
  }

  std::pair<int, int> succ(bool f, int l, int r, int level) {
    return {matrix[level].rank(f, l) + mid[level] * f,
            matrix[level].rank(f, r) + mid[level] * f};
  }

  // count d[i] s.t. (l <= i < r) && (v[i] < upper)
  D rect_sum(int l, int r, T upper) {
    D ret = 0;
    for (int level = MAXLOG - 1; level >= 0; level--) {
      bool f = ((upper >> level) & 1);
      if (f)
        ret += ds[level][matrix[level].rank(false, r)] -
               ds[level][matrix[level].rank(false, l)];
      std::tie(l, r) = succ(f, l, r, level);
    }
    return ret;
  }

  D rect_sum(int l, int r, T lower, T upper) {
    return rect_sum(l, r, upper) - rect_sum(l, r, lower);
  }
};

template <typename T, int MAXLOG, typename D>
struct CompressedWaveletMatrixRectangleSum {
  WaveletMatrixRectangleSum<int, MAXLOG, D> mat;
  std::vector<T> ys;

  CompressedWaveletMatrixRectangleSum(const std::vector<T>& v,
                                      const std::vector<D>& d)
      : ys(v) {
    std::sort(std::begin(ys), std::end(ys));
    ys.erase(std::unique(std::begin(ys), std::end(ys)), std::end(ys));
    std::vector<int> t(v.size());
    for (int i = 0; i < v.size(); i++) t[i] = get(v[i]);
    mat = WaveletMatrixRectangleSum<int, MAXLOG, D>(t, d);
  }

  inline int get(const T& x) {
    return std::lower_bound(std::begin(ys), std::end(ys), x) - std::begin(ys);
  }

  D rect_sum(int l, int r, T upper) { return mat.rect_sum(l, r, get(upper)); }

  D rect_sum(int l, int r, T lower, T upper) {
    return mat.rect_sum(l, r, get(lower), get(upper));
  }
};

#line 2 "structure/wavelet/wavelet-matrix-rectangle-sum.hpp"

#include <algorithm>
#include <cassert>
#include <cstddef>
#include <iterator>
#include <numeric>
#include <tuple>
#include <utility>
#include <vector>

#line 2 "structure/wavelet/succinct-indexable-dictionary.hpp"

#line 5 "structure/wavelet/succinct-indexable-dictionary.hpp"

/**
 * @brief Succinct Indexable Dictionary(完備辞書)
 */
struct SuccinctIndexableDictionary {
  std::size_t length;
  std::size_t blocks;
  std::vector<unsigned> bit, sum;

  SuccinctIndexableDictionary() = default;

  SuccinctIndexableDictionary(std::size_t length)
      : length(length), blocks((length + 31) >> 5) {
    bit.assign(blocks, 0U);
    sum.assign(blocks, 0U);
  }

  void set(int k) { bit[k >> 5] |= 1U << (k & 31); }

  void build() {
    sum[0] = 0U;
    for (int i = 1; i < blocks; i++) {
      sum[i] = sum[i - 1] + __builtin_popcount(bit[i - 1]);
    }
  }

  bool operator[](int k) { return (bool((bit[k >> 5] >> (k & 31)) & 1)); }

  int rank(int k) {
    return (sum[k >> 5] +
            __builtin_popcount(bit[k >> 5] & ((1U << (k & 31)) - 1)));
  }

  int rank(bool val, int k) { return (val ? rank(k) : k - rank(k)); }
};
#line 13 "structure/wavelet/wavelet-matrix-rectangle-sum.hpp"

/*
 * @brief Wavelet Matrix Rectangle Sum
 *
 */
template <typename T, int MAXLOG, typename D>
struct WaveletMatrixRectangleSum {
  std::size_t length;
  SuccinctIndexableDictionary matrix[MAXLOG];
  std::vector<D> ds[MAXLOG];
  int mid[MAXLOG];

  WaveletMatrixRectangleSum() = default;

  WaveletMatrixRectangleSum(const std::vector<T>& v, const std::vector<D>& d)
      : length(v.size()) {
    assert(v.size() == d.size());
    std::vector<int> l(length), r(length), ord(length);
    std::iota(std::begin(ord), std::end(ord), 0);
    for (int level = MAXLOG - 1; level >= 0; level--) {
      matrix[level] = SuccinctIndexableDictionary(length + 1);
      int left = 0, right = 0;
      for (int i = 0; i < length; i++) {
        if (((v[ord[i]] >> level) & 1)) {
          matrix[level].set(i);
          r[right++] = ord[i];
        } else {
          l[left++] = ord[i];
        }
      }
      mid[level] = left;
      matrix[level].build();
      std::swap(ord, l);
      for (int i = 0; i < right; i++) {
        ord[left + i] = r[i];
      }
      ds[level].resize(length + 1);
      ds[level][0] = D();
      for (int i = 0; i < length; i++) {
        ds[level][i + 1] = ds[level][i] + d[ord[i]];
      }
    }
  }

  std::pair<int, int> succ(bool f, int l, int r, int level) {
    return {matrix[level].rank(f, l) + mid[level] * f,
            matrix[level].rank(f, r) + mid[level] * f};
  }

  // count d[i] s.t. (l <= i < r) && (v[i] < upper)
  D rect_sum(int l, int r, T upper) {
    D ret = 0;
    for (int level = MAXLOG - 1; level >= 0; level--) {
      bool f = ((upper >> level) & 1);
      if (f)
        ret += ds[level][matrix[level].rank(false, r)] -
               ds[level][matrix[level].rank(false, l)];
      std::tie(l, r) = succ(f, l, r, level);
    }
    return ret;
  }

  D rect_sum(int l, int r, T lower, T upper) {
    return rect_sum(l, r, upper) - rect_sum(l, r, lower);
  }
};

template <typename T, int MAXLOG, typename D>
struct CompressedWaveletMatrixRectangleSum {
  WaveletMatrixRectangleSum<int, MAXLOG, D> mat;
  std::vector<T> ys;

  CompressedWaveletMatrixRectangleSum(const std::vector<T>& v,
                                      const std::vector<D>& d)
      : ys(v) {
    std::sort(std::begin(ys), std::end(ys));
    ys.erase(std::unique(std::begin(ys), std::end(ys)), std::end(ys));
    std::vector<int> t(v.size());
    for (int i = 0; i < v.size(); i++) t[i] = get(v[i]);
    mat = WaveletMatrixRectangleSum<int, MAXLOG, D>(t, d);
  }

  inline int get(const T& x) {
    return std::lower_bound(std::begin(ys), std::end(ys), x) - std::begin(ys);
  }

  D rect_sum(int l, int r, T upper) { return mat.rect_sum(l, r, get(upper)); }

  D rect_sum(int l, int r, T lower, T upper) {
    return mat.rect_sum(l, r, get(lower), get(upper));
  }
};