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#include "structure/wavelet/wavelet-matrix.hpp"
$2$ 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.
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.$V$ は値の最大値.
#include "succinct-indexable-dictionary.hpp"
/*
* @brief Wavelet Matrix(ウェーブレット行列)
*
*/
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)
*
*/
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];
}
};