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

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• Last update: 2024-05-26 19:28:30+09:00
• Include: #include "structure/wavelet/wavelet-matrix-rectangle-sum.hpp"

概要

$2$ 次元平面上にある重み付きの点が事前に与えられているとき, オンラインで矩形内にある点の重みの総和を効率的に求めるデータ構造.

普通のウェーブレット行列に対し重みの累積和を持たせた配列を用意すると, 矩形内の点の個数を数えるのと同じように重みの総和を求めることができる.

基本的には事前に高さを要素数に圧縮する CompressedWaveletMatrixRectangleSum を用いると高速に動作する.

使い方

• WaveletMatrixRectangleSum(v, d): 各要素の高さ v , 対応する要素の重み d を初期値として構築する.
• rect_sum(l, r, upper): 区間 $[l, r)$ の高さ $[0, upper)$ にある要素の重みの総和を返す.
• rect_sum(l, r, lower, upper): 区間 $[l, r)$ の高さ $[lower, upper)$ にある要素の重みの総和を返す.

計算量

• 構築: $O(N \log V)$
• クエリ: $O(\log V)$

$V$ は値の最大値.

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

#include "succinct-indexable-dictionary.hpp"

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

  WaveletMatrixRectangleSum() = default;

  WaveletMatrixRectangleSum(const vector<T> &v, const vector<D> &d)
      : length(v.size()) {
    assert(v.size() == d.size());
    vector<int> l(length), r(length), ord(length);
    iota(begin(ord), 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();
      ord.swap(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]];
      }
    }
  }

  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)];
      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;
  vector<T> ys;

  CompressedWaveletMatrixRectangleSum(const vector<T> &v, const vector<D> &d)
      : ys(v) {
    sort(begin(ys), end(ys));
    ys.erase(unique(begin(ys), end(ys)), end(ys));
    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 lower_bound(begin(ys), end(ys), x) - 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 1 "structure/wavelet/succinct-indexable-dictionary.hpp"
/**
 * @brief Succinct Indexable Dictionary(完備辞書)
 */
struct SuccinctIndexableDictionary {
  size_t length;
  size_t blocks;
  vector<unsigned> bit, sum;

  SuccinctIndexableDictionary() = default;

  SuccinctIndexableDictionary(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 2 "structure/wavelet/wavelet-matrix-rectangle-sum.hpp"

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

  WaveletMatrixRectangleSum() = default;

  WaveletMatrixRectangleSum(const vector<T> &v, const vector<D> &d)
      : length(v.size()) {
    assert(v.size() == d.size());
    vector<int> l(length), r(length), ord(length);
    iota(begin(ord), 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();
      ord.swap(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]];
      }
    }
  }

  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)];
      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;
  vector<T> ys;

  CompressedWaveletMatrixRectangleSum(const vector<T> &v, const vector<D> &d)
      : ys(v) {
    sort(begin(ys), end(ys));
    ys.erase(unique(begin(ys), end(ys)), end(ys));
    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 lower_bound(begin(ys), end(ys), x) - 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));
  }
};

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