Wavelet Matrix(ウェーブレット行列)
(structure/wavelet/wavelet-matrix.hpp)

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

概要

$2$ 次元平面上にある点が事前に与えられているとき, オンラインでいろいろなクエリを処理するデータ構造.

基本的には事前に要素の値を要素数に圧縮する CompressedWaveletMatrix を用いると高速に動作する.

使い方

• WaveletMatrix(v): 各要素の高さ v を初期値として構築する.
• access(k): $k$ 番目の要素を返す.
• rank(x, r): 区間 $[0, r)$ に含まれる x の個数を返す.
• kth_smallest(l, r, k): 区間 $[l, r)$ に含まれる要素のうち $k$ 番目(0-indexed) に小さいものを返す.
• kth_largest(l, r, k): 区間 $[l, r)$ に含まれる要素のうち $k$ 番目 (0-indexed) に大きいものを返す.
• range_freq(l, r, lower, upper): 区間 $[l, r)$ に含まれる要素のうち $[lower, upper)$ である要素数を返す.
• prev_value(l, r, upper): 区間 $[l, r)$ に含まれる要素のうち upper の次に小さいものを返す.
• next_value(l, r, lower): 区間 $[l, r)$ に含まれる要素のうち lower の次に大きいものを返す.

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/aoj-1549.test.cpp
• test/verify/aoj-2674.test.cpp
• test/verify/yosupo-range-kth-smallest.test.cpp

Code

#include "succinct-indexable-dictionary.hpp"

/*
 * @brief Wavelet Matrix(ウェーブレット行列)
 * @docs docs/wavelet-matrix.md
 */
template< typename T, int MAXLOG >
struct WaveletMatrix {
  size_t length;
  SuccinctIndexableDictionary matrix[MAXLOG];
  int mid[MAXLOG];

  WaveletMatrix() = default;

  WaveletMatrix(vector< T > v) : length(v.size()) {
    vector< T > l(length), r(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[i] >> level) & 1)) {
          matrix[level].set(i);
          r[right++] = v[i];
        } else {
          l[left++] = v[i];
        }
      }
      mid[level] = left;
      matrix[level].build();
      v.swap(l);
      for(int i = 0; i < right; i++) {
        v[left + i] = r[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};
  }

  // v[k]
  T access(int k) {
    T ret = 0;
    for(int level = MAXLOG - 1; level >= 0; level--) {
      bool f = matrix[level][k];
      if(f) ret |= T(1) << level;
      k = matrix[level].rank(f, k) + mid[level] * f;
    }
    return ret;
  }

  T operator[](const int &k) {
    return access(k);
  }

  // count i s.t. (0 <= i < r) && v[i] == x
  int rank(const T &x, int r) {
    int l = 0;
    for(int level = MAXLOG - 1; level >= 0; level--) {
      tie(l, r) = succ((x >> level) & 1, l, r, level);
    }
    return r - l;
  }

  // k-th(0-indexed) smallest number in v[l,r)
  T kth_smallest(int l, int r, int k) {
    assert(0 <= k && k < r - l);
    T ret = 0;
    for(int level = MAXLOG - 1; level >= 0; level--) {
      int cnt = matrix[level].rank(false, r) - matrix[level].rank(false, l);
      bool f = cnt <= k;
      if(f) {
        ret |= T(1) << level;
        k -= cnt;
      }
      tie(l, r) = succ(f, l, r, level);
    }
    return ret;
  }

  // k-th(0-indexed) largest number in v[l,r)
  T kth_largest(int l, int r, int k) {
    return kth_smallest(l, r, r - l - k - 1);
  }

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

  // count i s.t. (l <= i < r) && (lower <= v[i] < upper)
  int range_freq(int l, int r, T lower, T upper) {
    return range_freq(l, r, upper) - range_freq(l, r, lower);
  }

  // max v[i] s.t. (l <= i < r) && (v[i] < upper)
  T prev_value(int l, int r, T upper) {
    int cnt = range_freq(l, r, upper);
    return cnt == 0 ? T(-1) : kth_smallest(l, r, cnt - 1);
  }

  // min v[i] s.t. (l <= i < r) && (lower <= v[i])
  T next_value(int l, int r, T lower) {
    int cnt = range_freq(l, r, lower);
    return cnt == r - l ? T(-1) : kth_smallest(l, r, cnt);
  }
};

template< typename T, int MAXLOG >
struct CompressedWaveletMatrix {
  WaveletMatrix< int, MAXLOG > mat;
  vector< T > ys;

  CompressedWaveletMatrix(const vector< T > &v) : 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 = WaveletMatrix< int, MAXLOG >(t);
  }

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

  T access(int k) {
    return ys[mat.access(k)];
  }

  T operator[](const int &k) {
    return access(k);
  }

  int rank(const T &x, int r) {
    auto pos = get(x);
    if(pos == ys.size() || ys[pos] != x) return 0;
    return mat.rank(pos, r);
  }

  T kth_smallest(int l, int r, int k) {
    return ys[mat.kth_smallest(l, r, k)];
  }

  T kth_largest(int l, int r, int k) {
    return ys[mat.kth_largest(l, r, k)];
  }

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

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

  T prev_value(int l, int r, T upper) {
    auto ret = mat.prev_value(l, r, get(upper));
    return ret == -1 ? T(-1) : ys[ret];
  }

  T next_value(int l, int r, T lower) {
    auto ret = mat.next_value(l, r, get(lower));
    return ret == -1 ? T(-1) : ys[ret];
  }
};

#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.hpp"

/*
 * @brief Wavelet Matrix(ウェーブレット行列)
 * @docs docs/wavelet-matrix.md
 */
template< typename T, int MAXLOG >
struct WaveletMatrix {
  size_t length;
  SuccinctIndexableDictionary matrix[MAXLOG];
  int mid[MAXLOG];

  WaveletMatrix() = default;

  WaveletMatrix(vector< T > v) : length(v.size()) {
    vector< T > l(length), r(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[i] >> level) & 1)) {
          matrix[level].set(i);
          r[right++] = v[i];
        } else {
          l[left++] = v[i];
        }
      }
      mid[level] = left;
      matrix[level].build();
      v.swap(l);
      for(int i = 0; i < right; i++) {
        v[left + i] = r[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};
  }

  // v[k]
  T access(int k) {
    T ret = 0;
    for(int level = MAXLOG - 1; level >= 0; level--) {
      bool f = matrix[level][k];
      if(f) ret |= T(1) << level;
      k = matrix[level].rank(f, k) + mid[level] * f;
    }
    return ret;
  }

  T operator[](const int &k) {
    return access(k);
  }

  // count i s.t. (0 <= i < r) && v[i] == x
  int rank(const T &x, int r) {
    int l = 0;
    for(int level = MAXLOG - 1; level >= 0; level--) {
      tie(l, r) = succ((x >> level) & 1, l, r, level);
    }
    return r - l;
  }

  // k-th(0-indexed) smallest number in v[l,r)
  T kth_smallest(int l, int r, int k) {
    assert(0 <= k && k < r - l);
    T ret = 0;
    for(int level = MAXLOG - 1; level >= 0; level--) {
      int cnt = matrix[level].rank(false, r) - matrix[level].rank(false, l);
      bool f = cnt <= k;
      if(f) {
        ret |= T(1) << level;
        k -= cnt;
      }
      tie(l, r) = succ(f, l, r, level);
    }
    return ret;
  }

  // k-th(0-indexed) largest number in v[l,r)
  T kth_largest(int l, int r, int k) {
    return kth_smallest(l, r, r - l - k - 1);
  }

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

  // count i s.t. (l <= i < r) && (lower <= v[i] < upper)
  int range_freq(int l, int r, T lower, T upper) {
    return range_freq(l, r, upper) - range_freq(l, r, lower);
  }

  // max v[i] s.t. (l <= i < r) && (v[i] < upper)
  T prev_value(int l, int r, T upper) {
    int cnt = range_freq(l, r, upper);
    return cnt == 0 ? T(-1) : kth_smallest(l, r, cnt - 1);
  }

  // min v[i] s.t. (l <= i < r) && (lower <= v[i])
  T next_value(int l, int r, T lower) {
    int cnt = range_freq(l, r, lower);
    return cnt == r - l ? T(-1) : kth_smallest(l, r, cnt);
  }
};

template< typename T, int MAXLOG >
struct CompressedWaveletMatrix {
  WaveletMatrix< int, MAXLOG > mat;
  vector< T > ys;

  CompressedWaveletMatrix(const vector< T > &v) : 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 = WaveletMatrix< int, MAXLOG >(t);
  }

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

  T access(int k) {
    return ys[mat.access(k)];
  }

  T operator[](const int &k) {
    return access(k);
  }

  int rank(const T &x, int r) {
    auto pos = get(x);
    if(pos == ys.size() || ys[pos] != x) return 0;
    return mat.rank(pos, r);
  }

  T kth_smallest(int l, int r, int k) {
    return ys[mat.kth_smallest(l, r, k)];
  }

  T kth_largest(int l, int r, int k) {
    return ys[mat.kth_largest(l, r, k)];
  }

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

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

  T prev_value(int l, int r, T upper) {
    auto ret = mat.prev_value(l, r, get(upper));
    return ret == -1 ? T(-1) : ys[ret];
  }

  T next_value(int l, int r, T lower) {
    auto ret = mat.next_value(l, r, get(lower));
    return ret == -1 ? T(-1) : ys[ret];
  }
};

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