Wavelet Matrix Point Add Rectangle Sum
(structure/wavelet/wavelet-matrix-point-add-rectangle-sum.hpp)

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• Last update: 2022-09-11 00:53:50+09:00
• Include: #include "structure/wavelet/wavelet-matrix-point-add-rectangle-sum.hpp"

概要

$2$ 次元平面上にある点の位置が事前に与えられているとき, オンラインである点に対する重みの加算と, 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.

Wavelet-Matrix-Rectangle-Sum で用いた重みの累積和を持たせた配列をBinary-Indexed-Tree に置き換えると, ある点に対する重みの加算を効率的に行える.

基本的には事前に高さを要素数に圧縮する CompressedWaveletMatrixPointAddRectangleSum 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.
• point_add(k, x): 要素 $k$ の重みに $x$ を加算する.

計算量

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

$V$ は値の最大値.

Depends on

• Binary-Indexed-Tree(BIT) (structure/others/binary-indexed-tree.hpp)
• Succinct Indexable Dictionary(完備辞書) (structure/wavelet/succinct-indexable-dictionary.hpp)

Verified with

• test/verify/yosupo-point-add-rectangle-sum.test.cpp

Code

#include "succinct-indexable-dictionary.hpp"
#include "../others/binary-indexed-tree.hpp"

/*
 * @brief Wavelet Matrix Point Add Rectangle Sum
 * @docs docs/wavelet-matrix-point-add-rectangle-sum.md
 */
template< typename T, int MAXLOG, typename D >
struct WaveletMatrixPointAddRectangleSum {
  size_t length;
  SuccinctIndexableDictionary matrix[MAXLOG];
  BinaryIndexedTree< D > ds[MAXLOG];
  vector< T > v;
  int mid[MAXLOG];

  WaveletMatrixPointAddRectangleSum() = default;

  WaveletMatrixPointAddRectangleSum(const vector< T > &v, const vector< D > &d) : length(v.size()), v(v) {
    assert(v.size() == d.size());
    vector< int > l(length), r(length), ord(length);
    iota(begin(ord), end(ord), 0);
    vector< D > dd(length);
    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];
      }
      for(int i = 0; i < length; i++) {
        dd[i] = d[ord[i]];
      }
      ds[level] = BinaryIndexedTree< D >(dd);
    }
  }

  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--) {
      if(((upper >> level) & 1)) {
        auto nxt = succ(false, l, r, level);
        ret += ds[level].prod(nxt.first, nxt.second);
        l = l - nxt.first + mid[level];
        r = r - nxt.second + mid[level];
      } else {
        tie(l, r) = succ(false, 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);
  }

  void point_add(int k, const D &x) {
    auto &y = v[k];
    for(int level = MAXLOG - 1; level >= 0; level--) {
      bool f = ((y >> level) & 1);
      k = matrix[level].rank(f, k) + mid[level] * f;
      ds[level].apply(k, x);
    }
  }
};

template< typename T, int MAXLOG, typename D >
struct CompressedWaveletMatrixPointAddRectangleSum {
  WaveletMatrixPointAddRectangleSum< int, MAXLOG, D > mat;
  vector< T > ys;

  CompressedWaveletMatrixPointAddRectangleSum(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 = WaveletMatrixPointAddRectangleSum< 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));
  }

  void point_add(int k, const D &x) {
    mat.point_add(k, x);
  }
};

#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 1 "structure/others/binary-indexed-tree.hpp"
/**
 * @brief Binary-Indexed-Tree(BIT)
 * @docs docs/binary-indexed-tree.md
 */
template< typename T >
struct BinaryIndexedTree {
private:
  int n;
  vector< T > data;

public:
  BinaryIndexedTree() = default;

  explicit BinaryIndexedTree(int n) : n(n) {
    data.assign(n + 1, T());
  }

  explicit BinaryIndexedTree(const vector< T > &v) :
      BinaryIndexedTree((int) v.size()) {
    build(v);
  }

  void build(const vector< T > &v) {
    assert(n == (int) v.size());
    for(int i = 1; i <= n; i++) data[i] = v[i - 1];
    for(int i = 1; i <= n; i++) {
      int j = i + (i & -i);
      if(j <= n) data[j] += data[i];
    }
  }

  void apply(int k, const T &x) {
    for(++k; k <= n; k += k & -k) data[k] += x;
  }

  T prod(int r) const {
    T ret = T();
    for(; r > 0; r -= r & -r) ret += data[r];
    return ret;
  }

  T prod(int l, int r) const {
    return prod(r) - prod(l);
  }

  int lower_bound(T x) const {
    int i = 0;
    for(int k = 1 << (__lg(n) + 1); k > 0; k >>= 1) {
      if(i + k <= n && data[i + k] < x) {
        x -= data[i + k];
        i += k;
      }
    }
    return i;
  }

  int upper_bound(T x) const {
    int i = 0;
    for(int k = 1 << (__lg(n) + 1); k > 0; k >>= 1) {
      if(i + k <= n && data[i + k] <= x) {
        x -= data[i + k];
        i += k;
      }
    }
    return i;
  }
};
#line 3 "structure/wavelet/wavelet-matrix-point-add-rectangle-sum.hpp"

/*
 * @brief Wavelet Matrix Point Add Rectangle Sum
 * @docs docs/wavelet-matrix-point-add-rectangle-sum.md
 */
template< typename T, int MAXLOG, typename D >
struct WaveletMatrixPointAddRectangleSum {
  size_t length;
  SuccinctIndexableDictionary matrix[MAXLOG];
  BinaryIndexedTree< D > ds[MAXLOG];
  vector< T > v;
  int mid[MAXLOG];

  WaveletMatrixPointAddRectangleSum() = default;

  WaveletMatrixPointAddRectangleSum(const vector< T > &v, const vector< D > &d) : length(v.size()), v(v) {
    assert(v.size() == d.size());
    vector< int > l(length), r(length), ord(length);
    iota(begin(ord), end(ord), 0);
    vector< D > dd(length);
    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];
      }
      for(int i = 0; i < length; i++) {
        dd[i] = d[ord[i]];
      }
      ds[level] = BinaryIndexedTree< D >(dd);
    }
  }

  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--) {
      if(((upper >> level) & 1)) {
        auto nxt = succ(false, l, r, level);
        ret += ds[level].prod(nxt.first, nxt.second);
        l = l - nxt.first + mid[level];
        r = r - nxt.second + mid[level];
      } else {
        tie(l, r) = succ(false, 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);
  }

  void point_add(int k, const D &x) {
    auto &y = v[k];
    for(int level = MAXLOG - 1; level >= 0; level--) {
      bool f = ((y >> level) & 1);
      k = matrix[level].rank(f, k) + mid[level] * f;
      ds[level].apply(k, x);
    }
  }
};

template< typename T, int MAXLOG, typename D >
struct CompressedWaveletMatrixPointAddRectangleSum {
  WaveletMatrixPointAddRectangleSum< int, MAXLOG, D > mat;
  vector< T > ys;

  CompressedWaveletMatrixPointAddRectangleSum(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 = WaveletMatrixPointAddRectangleSum< 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));
  }

  void point_add(int k, const D &x) {
    mat.point_add(k, x);
  }
};

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