This documentation is automatically generated by competitive-verifier/competitive-verifier
#include "structure/wavelet/wavelet-tree.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 CompressedWaveletTree を用いると高速に動作する.
ウェーブレット行列を用いたほうが時間と空間の計算量が良いため, 使い所なし. 悲しい.
WaveletTree(v)
: 各要素の高さ v
を初期値として構築する.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.
/*
* @brief Wavelet Tree(ウェーブレット木)
*
*/
template <typename T, int MAXLOG>
struct WaveletTree {
struct Node {
SuccinctIndexableDictionary sid;
Node *ch[2];
Node() = default;
Node(size_t length) : sid(length + 1), ch{nullptr} {}
};
Node *root;
Node *build(vector<T> &v, vector<T> &rbuff, int bit, int l, int r) {
if (l >= r || bit == -1) return nullptr;
Node *node = new Node(r - l);
int left = 0, right = 0;
for (int k = l; k < r; k++) {
if (((v[k] >> bit) & 1)) {
rbuff[right++] = v[k];
node->sid.set(k - l);
} else {
v[l + left++] = v[k];
}
}
for (int k = 0; k < right; k++) {
v[l + left + k] = rbuff[k];
}
node->sid.build();
node->ch[0] = build(v, rbuff, bit - 1, l, l + left);
node->ch[1] = build(v, rbuff, bit - 1, l + left, r);
return node;
}
WaveletTree() = default;
WaveletTree(vector<T> v) {
vector<T> rbuff(v.size());
root = build(v, rbuff, MAXLOG - 1, 0, v.size());
}
int rank(Node *t, int l, int r, const T &x, int level) {
if (l >= r || t == nullptr) return 0;
if (level == -1) return r - l;
bool f = (x >> level) & 1;
l = t->sid.rank(f, l), r = t->sid.rank(f, r);
return rank(t->ch[f], l, r, x, level - 1);
}
int rank(const T &x, int r) { return rank(root, 0, r, x, MAXLOG - 1); }
T kth_smallest(Node *t, int l, int r, int k, int level) {
if (l >= r || t == nullptr) return 0;
int cnt = t->sid.rank(false, r) - t->sid.rank(false, l);
bool f = cnt <= k;
l = t->sid.rank(f, l), r = t->sid.rank(f, r);
if (f)
return kth_smallest(t->ch[f], l, r, k - cnt, level - 1) |
((T(1) << level));
return kth_smallest(t->ch[f], l, r, k, level - 1);
}
// 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);
return kth_smallest(root, l, r, k, MAXLOG - 1);
}
// 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);
}
int range_freq(Node *t, int l, int r, T upper, int level) {
if (t == nullptr || l >= r) return 0;
bool f = ((upper >> level) & 1);
int ret = 0;
if (f) ret += t->sid.rank(false, r) - t->sid.rank(false, l);
l = t->sid.rank(f, l), r = t->sid.rank(f, r);
return range_freq(t->ch[f], l, r, upper, level - 1) + ret;
}
// count i s.t. (l <= i < r) && (v[i] < upper)
int range_freq(int l, int r, T upper) {
return range_freq(root, l, r, upper, MAXLOG - 1);
}
// 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 CompressedWaveletTree {
WaveletTree<int, MAXLOG> mat;
vector<T> ys;
CompressedWaveletTree(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 = WaveletTree<int, MAXLOG>(t);
}
inline int get(const T &x) {
return lower_bound(begin(ys), end(ys), x) - begin(ys);
}
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/wavelet-tree.hpp"
/*
* @brief Wavelet Tree(ウェーブレット木)
*
*/
template <typename T, int MAXLOG>
struct WaveletTree {
struct Node {
SuccinctIndexableDictionary sid;
Node *ch[2];
Node() = default;
Node(size_t length) : sid(length + 1), ch{nullptr} {}
};
Node *root;
Node *build(vector<T> &v, vector<T> &rbuff, int bit, int l, int r) {
if (l >= r || bit == -1) return nullptr;
Node *node = new Node(r - l);
int left = 0, right = 0;
for (int k = l; k < r; k++) {
if (((v[k] >> bit) & 1)) {
rbuff[right++] = v[k];
node->sid.set(k - l);
} else {
v[l + left++] = v[k];
}
}
for (int k = 0; k < right; k++) {
v[l + left + k] = rbuff[k];
}
node->sid.build();
node->ch[0] = build(v, rbuff, bit - 1, l, l + left);
node->ch[1] = build(v, rbuff, bit - 1, l + left, r);
return node;
}
WaveletTree() = default;
WaveletTree(vector<T> v) {
vector<T> rbuff(v.size());
root = build(v, rbuff, MAXLOG - 1, 0, v.size());
}
int rank(Node *t, int l, int r, const T &x, int level) {
if (l >= r || t == nullptr) return 0;
if (level == -1) return r - l;
bool f = (x >> level) & 1;
l = t->sid.rank(f, l), r = t->sid.rank(f, r);
return rank(t->ch[f], l, r, x, level - 1);
}
int rank(const T &x, int r) { return rank(root, 0, r, x, MAXLOG - 1); }
T kth_smallest(Node *t, int l, int r, int k, int level) {
if (l >= r || t == nullptr) return 0;
int cnt = t->sid.rank(false, r) - t->sid.rank(false, l);
bool f = cnt <= k;
l = t->sid.rank(f, l), r = t->sid.rank(f, r);
if (f)
return kth_smallest(t->ch[f], l, r, k - cnt, level - 1) |
((T(1) << level));
return kth_smallest(t->ch[f], l, r, k, level - 1);
}
// 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);
return kth_smallest(root, l, r, k, MAXLOG - 1);
}
// 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);
}
int range_freq(Node *t, int l, int r, T upper, int level) {
if (t == nullptr || l >= r) return 0;
bool f = ((upper >> level) & 1);
int ret = 0;
if (f) ret += t->sid.rank(false, r) - t->sid.rank(false, l);
l = t->sid.rank(f, l), r = t->sid.rank(f, r);
return range_freq(t->ch[f], l, r, upper, level - 1) + ret;
}
// count i s.t. (l <= i < r) && (v[i] < upper)
int range_freq(int l, int r, T upper) {
return range_freq(root, l, r, upper, MAXLOG - 1);
}
// 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 CompressedWaveletTree {
WaveletTree<int, MAXLOG> mat;
vector<T> ys;
CompressedWaveletTree(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 = WaveletTree<int, MAXLOG>(t);
}
inline int get(const T &x) {
return lower_bound(begin(ys), end(ys), x) - begin(ys);
}
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];
}
};