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graph/ds/tree_wavelet_matrix.hpp
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- Last update: 2026-09-13 16:24:01+09:00
- Include:
#include "graph/ds/tree_wavelet_matrix.hpp" - Link: https://atcoder.jp/contests/pakencamp-2022-day1/tasks/pakencamp_2022_day1_j
- Link: https://atcoder.jp/contests/utpc2011/tasks/utpc2011_12
Depends on
alg/monoid/dummy.hpp
- ds/bit_vector.hpp
- ds/dummy_data_structure.hpp
- ds/hashmap.hpp
- ds/wavelet_matrix/wavelet_matrix.hpp
- graph/base.hpp
- graph/tree.hpp
- other/bit.hpp
Code
#include "ds/wavelet_matrix/wavelet_matrix.hpp"
#include "graph/tree.hpp"
// https://atcoder.jp/contests/pakencamp-2022-day1/tasks/pakencamp_2022_day1_j
// https://atcoder.jp/contests/utpc2011/tasks/utpc2011_12
template <typename TREE, bool edge, typename T, bool COMPRESS,
typename Monoid = Monoid_Add<T>>
struct Tree_Wavelet_Matrix {
TREE& tree;
int N;
using WM = Wavelet_Matrix<T, COMPRESS, Monoid_Add<T>>;
using X = typename Monoid::value_type;
WM wm;
Tree_Wavelet_Matrix(TREE& tree, vc<T> A, vc<X> SUM_data = {}, int log = -1)
: tree(tree), N(tree.N) {
vc<X>& S = SUM_data;
vc<T> A1;
vc<X> S1;
A1.resize(N);
if (!S.empty()) S1.resize(N);
if (!edge) {
assert(len(A) == N && (len(S) == 0 || len(S) == N));
FOR(v, N) A1[tree.LID[v]] = A[v];
if (len(S) == N) { FOR(v, N) S1[tree.LID[v]] = S[v]; }
wm.build(A1, S1, log);
} else {
assert(len(A) == N - 1 && (len(S) == 0 || len(S) == N - 1));
if (!S.empty()) {
FOR(e, N - 1) { S1[tree.LID[tree.e_to_v(e)]] = S[e]; }
}
FOR(e, N - 1) { A1[tree.LID[tree.e_to_v(e)]] = A[e]; }
wm.build(A1, S1, log);
}
}
// xor した結果で [a, b) に収まるものを数える
int count_path(int s, int t, T a, T b, T xor_val = 0) {
return wm.count(get_segments(s, t), a, b, xor_val);
}
// xor した結果で [a, b) に収まるものを数える
int count_subtree(int u, T a, T b, T xor_val = 0) {
int l = tree.LID[u], r = tree.RID[u];
return wm.count(l + edge, r, a, b, xor_val);
}
// xor した結果で、[L, R) の中で k>=0 番目と prefix sum
pair<T, X> kth_value_and_sum_path(int s, int t, int k, T xor_val = 0) {
return wm.kth_value_and_sum(get_segments(s, t), k, xor_val);
}
// xor した結果で、[L, R) の中で k>=0 番目と prefix sum
pair<T, X> kth_value_and_sum_subtree(int u, int k, T xor_val = 0) {
int l = tree.LID[u], r = tree.RID[u];
return wm.kth_value_and_sum(l + edge, r, k, xor_val);
}
// xor した結果で、[L, R) の中で k>=0 番目
T kth_path(int s, int t, int k, T xor_val = 0) {
return wm.kth(get_segments(s, t), k, xor_val);
}
// xor した結果で、[L, R) の中で k>=0 番目
T kth_subtree(int u, int k, T xor_val = 0) {
int l = tree.LID[u], r = tree.RID[u];
return wm.kth(l + edge, r, k, xor_val);
}
// xor した結果で、[L, R) の中で中央値。
// LOWER = true：下側中央値、false：上側中央値
T median_path(bool UPPER, int s, int t, T xor_val = 0) {
return wm.median(UPPER, get_segments(s, t), xor_val);
}
T median_subtree(bool UPPER, int u, T xor_val = 0) {
int l = tree.LID[u], r = tree.RID[u];
return wm.median(UPPER, l + edge, r, xor_val);
}
// xor した結果で [k1, k2) 番目であるところの SUM_data の和
X sum_path(int s, int t, int k1, int k2, T xor_val = 0) {
return wm.sum(get_segments(s, t), k1, k2, xor_val);
}
// xor した結果で [k1, k2) 番目であるところの SUM_data の和
X sum_subtree(int u, int k1, int k2, T xor_val = 0) {
int l = tree.LID[u], r = tree.RID[u];
return wm.sum(l + edge, r, k1, k2, xor_val);
}
X sum_all_path(int s, int t) { return wm.sum_all(get_segments(s, t)); }
X sum_all_subtree(int u) {
int l = tree.LID[u], r = tree.RID[u];
return wm.sum_all(l + edge, r);
}
private:
vc<pair<int, int>> get_segments(int s, int t) {
vc<pair<int, int>> segments = tree.get_path_decomposition(s, t, edge);
for (auto&& [a, b]: segments) {
if (a >= b) swap(a, b);
++b;
}
return segments;
}
};#line 1 "graph/ds/tree_wavelet_matrix.hpp"
#line 1 "other/bit.hpp"
int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_sgn(int x) { return (__builtin_parity(unsigned(x)) & 1 ? -1 : 1); }
int popcnt_sgn(u32 x) { return (__builtin_parity(x) & 1 ? -1 : 1); }
int popcnt_sgn(ll x) { return (__builtin_parityll(x) & 1 ? -1 : 1); }
int popcnt_sgn(u64 x) { return (__builtin_parityll(x) & 1 ? -1 : 1); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
template <typename T>
T kth_bit(int k) {
assert(0 <= k && k < int(8 * sizeof(T)));
return T(1) << k;
}
template <typename T>
bool has_kth_bit(T x, int k) {
assert(0 <= k && k < int(8 * sizeof(T)));
return x >> k & 1;
}
template <typename UINT>
struct all_bit {
static_assert(is_unsigned<UINT>::value);
UINT s;
all_bit(UINT s) : s(s) {}
struct iter {
UINT s;
int operator*() const { return lowbit(s); }
void operator++() { s &= s - 1; }
bool operator!=(nullptr_t) const { return s; }
};
iter begin() const { return {s}; }
nullptr_t end() const { return nullptr; }
};
template <typename UINT>
struct all_subset {
static_assert(is_unsigned<UINT>::value);
UINT s;
all_subset(UINT s) : s(s) {}
struct iter {
UINT s, t;
bool done = false;
UINT operator*() const { return t; }
void operator++() {
done = (t == 0);
t = (t - 1) & s;
}
bool operator!=(nullptr_t) const { return !done; }
};
iter begin() const { return {s, s}; }
nullptr_t end() const { return nullptr; }
};
constexpr u64 full_mask(int n) {
assert(0 <= n && n <= 64);
return n == 64 ? -1ULL : (1ULL << n) - 1;
}
u64 bit_reverse(u64 x) {
x = ((x & 0x5555555555555555ULL) << 1) | ((x >> 1) & 0x5555555555555555ULL);
x = ((x & 0x3333333333333333ULL) << 2) | ((x >> 2) & 0x3333333333333333ULL);
x = ((x & 0x0f0f0f0f0f0f0f0fULL) << 4) | ((x >> 4) & 0x0f0f0f0f0f0f0f0fULL);
x = ((x & 0x00ff00ff00ff00ffULL) << 8) | ((x >> 8) & 0x00ff00ff00ff00ffULL);
x = ((x & 0x0000ffff0000ffffULL) << 16) | ((x >> 16) & 0x0000ffff0000ffffULL);
x = (x << 32) | (x >> 32);
return x;
}
#line 2 "ds/bit_vector.hpp"
struct Bit_Vector {
int n;
bool prepared = 0;
vc<pair<u64, u32>> dat;
Bit_Vector(int n = 0) : n(n) { dat.assign((n + 127) >> 6, {0, 0}); }
void set(int i) {
assert(!prepared && (0 <= i && i < n));
dat[i >> 6].fi |= u64(1) << (i & 63);
}
void reset() {
fill(all(dat), pair<u64, u32>{0, 0});
prepared = 0;
}
void build() {
prepared = 1;
FOR(i, len(dat) - 1) dat[i + 1].se = dat[i].se + popcnt(dat[i].fi);
}
bool operator[](int i) const { return dat[i >> 6].fi >> (i & 63) & 1; }
// [0, k) 内の 1 の個数
int count_prefix(int k, bool f = true) const {
assert(prepared);
auto [a, b] = dat[k >> 6];
int ret = b + popcnt(a & ((u64(1) << (k & 63)) - 1));
return (f ? ret : k - ret);
}
int count(int L, int R, bool f = true) const {
return count_prefix(R, f) - count_prefix(L, f);
}
string to_string() const {
string ans;
FOR(i, n) ans += '0' + (dat[i / 64].fi >> (i % 64) & 1);
return ans;
}
};
#line 1 "alg/monoid/dummy.hpp"
struct Monoid_Dummy {
using value_type = char;
static constexpr bool commute = true;
static value_type op(value_type, value_type) { return 0; }
static value_type id() { return 0; }
};
#line 2 "ds/dummy_data_structure.hpp"
struct Dummy_Data_Structure {
using MX = Monoid_Dummy;
using T = typename MX::value_type;
void build(const vc<T>& A) {}
};
#line 3 "ds/wavelet_matrix/wavelet_matrix.hpp"
template <typename Y, typename SEGTREE>
struct Uncompressed_Wavelet_Matrix {
using Mono = typename SEGTREE::MX;
using T = typename Mono::value_type;
static_assert(Mono::commute);
static_assert(is_same_v<Y, int> || is_same_v<Y, ll>);
int n = 0, log = 0;
vc<int> mid;
vc<Bit_Vector> bv;
vc<SEGTREE> seg;
Y limit;
Uncompressed_Wavelet_Matrix() = default;
// f(i) = {A[i], dat[i]}
template <typename F>
Uncompressed_Wavelet_Matrix(int n, F f, int log = -1) {
build(n, f, log);
}
Uncompressed_Wavelet_Matrix(const vc<Y>& A, int log = -1) {
static_assert(is_same_v<SEGTREE, Dummy_Data_Structure>);
build(len(A), [&](int i) -> pair<Y, T> { return {A[i], Mono::id()}; }, log);
}
template <typename F>
void build(int n, F f, int log = -1) {
this->n = n;
vc<Y> A(n);
vc<T> S(n);
FOR(i, n) tie(A[i], S[i]) = f(i);
if (log == -1) {
log = (n == 0 ? 0 : topbit(MAX(A)) + 1);
} else {
for (auto& x : A) assert(0 <= x && topbit(x) < log);
}
this->log = log;
limit = Y(1) << log;
if constexpr (is_same_v<Y, int>) assert(0 <= log && log <= 30);
if constexpr (is_same_v<Y, ll>) assert(0 <= log && log <= 62);
mid.resize(log), bv.assign(log, Bit_Vector(n));
vc<Y> A0(n), A1(n);
vc<T> S0(n), S1(n);
seg.resize(log + 1);
seg[log].build(S);
for (int d = log - 1; d >= 0; --d) {
int p0 = 0, p1 = 0;
for (int i = 0; i < n; ++i) {
if (A[i] >> d & 1) {
bv[d].set(i), A1[p1] = A[i], S1[p1] = S[i], p1++;
} else {
A0[p0] = A[i], S0[p0] = S[i], p0++;
}
}
swap(A, A0), swap(S, S0);
move(A1.begin(), A1.begin() + p1, A.begin() + p0);
move(S1.begin(), S1.begin() + p1, S.begin() + p0);
mid[d] = p0, bv[d].build(), seg[d].build(S);
}
}
tuple<int, int, int, int> get_subtree(int d, int L, int R) const {
assert(1 <= d && d <= log);
int a = bv[d - 1].count_prefix(L), b = bv[d - 1].count_prefix(R);
return {L - a, R - b, mid[d - 1] + a, mid[d - 1] + b};
}
template <typename F>
void work_point(F f, int i) {
assert(0 <= i && i < n);
f(log, i);
FOR_R(d, log) {
int a = bv[d].count_prefix(i);
if (bv[d][i]) {
i = mid[d] + a;
} else {
i = i - a;
}
f(d, i);
}
}
template <typename F>
void work_prefix(F f, int L, int R, Y y) const {
chmin(y, limit);
if (y == 0) return;
if (y == limit) {
f(log, L, R);
return;
}
FOR_R(d, log) {
auto [L0, R0, L1, R1] = get_subtree(d + 1, L, R);
if (y >> d & 1) {
f(d, L0, R0);
L = L1, R = R1;
} else {
L = L0, R = R0;
}
}
}
template <typename F>
void work_range(F f, int L, int R, Y y1, Y y2) const {
chmin(y2, limit);
if (y1 >= y2) return;
assert(0 <= y1 && y1 <= y2 && y2 <= limit);
if (y1 == 0) return work_prefix(f, L, R, y2);
auto dfs = [&](auto& dfs, int d, int L, int R, Y y1, Y y2) -> void {
if (y1 == y2) return;
if (y1 == 0 && y2 == Y(1) << d) {
f(d, L, R);
return;
}
assert(d > 0);
auto [L0, R0, L1, R1] = get_subtree(d, L, R);
Y m = (Y(1) << (d - 1));
if (y2 <= m) {
dfs(dfs, d - 1, L0, R0, y1, y2);
} else if (y1 >= m) {
dfs(dfs, d - 1, L1, R1, y1 - m, y2 - m);
} else {
dfs(dfs, d - 1, L0, R0, y1, m);
dfs(dfs, d - 1, L1, R1, 0, y2 - m);
}
};
dfs(dfs, log, L, R, y1, y2);
}
// [L,R) x [0,y)
int prefix_count(int L, int R, Y y) const {
int cnt = 0;
work_prefix([&](int d, int a, int b) { cnt += b - a; }, L, R, y);
return cnt;
}
// [L,R) x [y1,y2)
int count(int L, int R, Y y1, Y y2) const {
return prefix_count(L, R, y2) - prefix_count(L, R, y1);
}
// [L,R) x [0,y)
T prefix_prod(int L, int R, Y y) const {
T ans = Mono::id();
work_prefix(
[&](int d, int a, int b) { ans = Mono::op(ans, seg[d].prod(a, b)); }, L,
R, y);
return ans;
}
// [L,R) x [y1,y2)
T prod(int L, int R, Y y1, Y y2) const {
T ans = Mono::id();
work_range(
[&](int d, int a, int b) { ans = Mono::op(ans, seg[d].prod(a, b)); }, L,
R, y1, y2);
return ans;
}
T prod_all(int L, int R) const { return seg[log].prod(L, R); }
// [L,R) x [0,y)
pair<int, T> prefix_count_and_prod(int L, int R, Y y) const {
pair<int, T> ans = {0, Mono::id()};
work_prefix(
[&](int d, int a, int b) {
ans.fi += b - a;
ans.se = Mono::op(ans.se, seg[d].prod(a, b));
},
L, R, y);
return ans;
}
// [L,R) x [y1,y2)
pair<int, T> count_and_prod(int L, int R, Y y1, Y y2) const {
pair<int, T> ans = {0, Mono::id()};
work_range(
[&](int d, int a, int b) {
ans.fi += b - a;
ans.se = Mono::op(ans.se, seg[d].prod(a, b));
},
L, R, y1, y2);
return ans;
}
Y kth(int L, int R, int k) const {
assert(0 <= k && k < R - L);
Y ans = 0;
for (int d = log - 1; d >= 0; --d) {
auto [L0, R0, L1, R1] = get_subtree(d + 1, L, R);
if (k < R0 - L0) {
L = L0, R = R0;
} else {
ans |= Y(1) << d;
k -= R0 - L0, L = L1, R = R1;
}
}
return ans;
}
template <bool upper>
Y median(int L, int R) const {
assert(0 <= L && L < R && R <= n);
int k = (upper ? (R - L) / 2 : (R - L - 1) / 2);
return kth(L, R, k);
}
void set(int i, T t) {
assert(0 <= i && i < n);
work_point([&](int d, int i) { seg[d].set(i, t); }, i);
}
void multiply(int i, T t) {
assert(0 <= i && i < n);
work_point([&](int d, int i) { seg[d].multiply(i, t); }, i);
}
void add(int i, T t) {
assert(0 <= i && i < n);
work_point([&](int d, int i) { seg[d].add(i, t); }, i);
}
// [L,R) x [0,y) での check(y, cnt, prod) が true となる最大の (Y,cnt,prod)
// cnt はデータ件数全体であって, activate/deactivate を考慮する場合には
// prod の方を見る必要がある
template <typename F>
tuple<Y, int, T> max_right(F check, int L, int R) const {
int cnt = 0;
Y y = 0;
T t = Mono::id();
T t_all = seg[log].prod(L, R);
assert(check(0, 0, Mono::id()));
if (check(infty<Y>, R - L, t_all)) {
return {infty<Y>, R - L, t_all};
}
for (int d = log - 1; d >= 0; --d) {
auto [L0, R0, L1, R1] = get_subtree(d + 1, L, R);
Y y1 = y | Y(1) << d;
int cnt1 = cnt + R0 - L0;
T t1 = Mono::op(t, seg[d].prod(L0, R0));
if (check(y1, cnt1, t1)) {
y = y1, cnt = cnt1, t = t1, L = L1, R = R1;
} else {
L = L0, R = R0;
}
}
return {y, cnt, t};
}
// [L,R) x [0,y) での check(y, cnt, prod) が true となる最大の (Y,cnt,prod)
template <typename F>
tuple<Y, int, T> max_right_many(F check, vc<pair<int, int>> LR) const {
int cnt = 0;
Y y = 0;
T t = Mono::id();
T t_all = Mono::id();
int cnt_all = 0;
for (auto& [l, r] : LR)
t_all = Mono::op(t_all, prod_all(l, r)), cnt_all += r - l;
assert(check(0, 0, Mono::id()));
if (check(infty<Y>, cnt_all, t_all)) {
return {infty<Y>, cnt_all, t_all};
}
for (int d = log - 1; d >= 0; --d) {
Y y1 = Y(1) << d;
T t1 = t;
int cnt1 = 0;
for (auto& [L, R] : LR) {
auto [L0, R0, L1, R1] = get_subtree(d + 1, L, R);
cnt1 += R0 - L0;
t1 = Mono::op(t1, seg[d].prod(L0, R0));
}
if (check(y1, cnt1, t1)) {
y = y1, cnt = cnt1, t = t1;
for (auto& [L, R] : LR) {
auto [L0, R0, L1, R1] = get_subtree(d + 1, L, R);
L = L1, R = R1;
}
} else {
for (auto& [L, R] : LR) {
auto [L0, R0, L1, R1] = get_subtree(d + 1, L, R);
L = L0, R = R0;
}
}
}
return {y, cnt, t};
}
// [L,R) x [y, inf) での check(y, cnt, prod) が true となる最小の (y,cnt,prod)
// cnt==0 だと true であることは仮定する
// https://qoj.ac/contest/1047/problem/5094
template <typename F>
tuple<Y, int, T> min_left_many(F check, vc<pair<int, int>> LR) const {
assert(check(limit, 0, Mono::id()));
int cnt = 0;
Y y = limit;
T t = Mono::id();
T t_all = Mono::id();
int cnt_all = 0;
for (auto& [l, r] : LR)
t_all = Mono::op(t_all, prod_all(l, r)), cnt_all += r - l;
if (check(0, cnt_all, t_all)) {
return {0, cnt_all, t_all};
}
for (int d = log - 1; d >= 0; --d) {
Y y1 = y - (Y(1) << d);
T t1 = t;
int cnt1 = cnt;
for (auto& [L, R] : LR) {
auto [L0, R0, L1, R1] = get_subtree(d + 1, L, R);
cnt1 += R1 - L1;
t1 = Mono::op(t1, seg[d].prod(L1, R1));
}
if (check(y1, cnt1, t1)) {
y = y1, cnt = cnt1, t = t1;
SHOW(y);
for (auto& [L, R] : LR) {
auto [L0, R0, L1, R1] = get_subtree(d + 1, L, R);
L = L0, R = R0;
}
} else {
for (auto& [L, R] : LR) {
auto [L0, R0, L1, R1] = get_subtree(d + 1, L, R);
L = L1, R = R1;
}
}
}
SHOW(y, cnt, t);
return {y, cnt, t};
}
};
template <typename Y, typename SEGTREE>
struct Compressed_Wavelet_Matrix {
using Mono = typename SEGTREE::MX;
using T = typename Mono::value_type;
int n = 0;
vc<Y> key;
Uncompressed_Wavelet_Matrix<int, SEGTREE> wm;
Compressed_Wavelet_Matrix() = default;
// f(i) = {A[i], dat[i]}
template <typename F>
Compressed_Wavelet_Matrix(int n, F f) {
build(n, f);
}
Compressed_Wavelet_Matrix(const vc<Y>& A) {
static_assert(is_same_v<SEGTREE, Dummy_Data_Structure>);
build(A);
}
template <typename F>
void build(int n, F f) {
this->n = n;
vc<Y> A(n);
vc<T> S(n);
FOR(i, n) tie(A[i], S[i]) = f(i);
key = A;
UNIQUE(key);
wm.build(n, [&](int i) -> pair<int, T> {
int k = LB(key, A[i]);
return {k, S[i]};
});
}
void build(const vc<Y>& A) {
static_assert(is_same_v<SEGTREE, Dummy_Data_Structure>);
n = len(A);
key = A;
UNIQUE(key);
wm.build(n, [&](int i) -> pair<int, T> {
int k = LB(key, A[i]);
return {k, Mono::id()};
});
}
Y kth(int L, int R, int k) const { return key[wm.kth(L, R, k)]; }
template <bool upper>
Y median(int L, int R) const {
return key[wm.template median<upper>(L, R)];
}
// [L,R) x [-inf,y)
int prefix_count(int L, int R, Y y) const {
return wm.prefix_count(L, R, LB(key, y));
}
// [L,R) x [y1,y2)
int count(int L, int R, Y y1, Y y2) const {
return wm.count(L, R, LB(key, y1), LB(key, y2));
}
// [L,R) x [-inf,y)
T prefix_prod(int L, int R, Y y) const {
return wm.prefix_prod(L, R, LB(key, y));
}
// [L,R) x [y1,y2)
T prod(int L, int R, Y y1, Y y2) const {
return wm.prod(L, R, LB(key, y1), LB(key, y2));
}
T prod_all(int L, int R) const { return wm.prod_all(L, R); }
// [L,R) x [-inf,y)
pair<int, T> prefix_count_and_prod(int L, int R, Y y) const {
return wm.prefix_count_and_prod(L, R, LB(key, y));
}
// [L,R) x [y1,y2)
pair<int, T> count_and_prod(int L, int R, Y y1, Y y2) const {
return wm.count_and_prod(L, R, LB(key, y1), LB(key, y2));
}
void set(int i, T t) { wm.set(i, t); }
void multiply(int i, T t) { wm.multiply(i, t); }
void add(int i, T t) { wm.add(i, t); }
};
template <typename Y, bool compress, typename SEGTREE = Dummy_Data_Structure>
using Wavelet_Matrix =
conditional_t<compress, Compressed_Wavelet_Matrix<Y, SEGTREE>,
Uncompressed_Wavelet_Matrix<Y, SEGTREE>>;
#line 1 "graph/tree.hpp"
#line 1 "ds/hashmap.hpp"
// u64 -> Val
template <typename Val>
struct HashMap {
// n は入れたいものの個数で ok
HashMap(u32 n = 0) { build(n); }
void build(u32 n) {
u32 k = 8;
while (k < n * 2) k *= 2;
cap = k / 2, mask = k - 1;
key.resize(k), val.resize(k), used.assign(k, 0);
}
// size を保ったまま. size=0 にするときは build すること.
void clear() {
used.assign(len(used), 0);
cap = (mask + 1) / 2;
}
int size() { return len(used) / 2 - cap; }
int index(const u64& k) {
int i = 0;
for (i = hash(k); used[i] && key[i] != k; i = (i + 1) & mask) {
}
return i;
}
Val& operator[](const u64& k) {
int i = index(k);
if (used[i]) return val[i];
if (cap == 0) extend(), i = index(k);
used[i] = 1, key[i] = k, val[i] = Val{}, --cap;
return val[i];
}
Val get(const u64& k, Val default_value) {
int i = index(k);
return (used[i] ? val[i] : default_value);
}
bool count(const u64& k) {
int i = index(k);
return used[i] && key[i] == k;
}
// f(key, val)
template <typename F>
void enumerate_all(F f) {
FOR(i, len(used)) if (used[i]) f(key[i], val[i]);
}
private:
u32 cap, mask;
vc<u64> key;
vc<Val> val;
vc<bool> used;
u64 hash(u64 x) {
static const u64 FIXED_RANDOM =
std::chrono::steady_clock::now().time_since_epoch().count();
x += FIXED_RANDOM;
x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
return (x ^ (x >> 31)) & mask;
}
void extend() {
vc<pair<u64, Val>> dat;
dat.reserve(len(used) / 2 - cap);
FOR(i, len(used)) {
if (used[i]) dat.eb(key[i], val[i]);
}
build(2 * len(dat));
for (auto& [a, b] : dat) (*this)[a] = b;
}
};
#line 2 "graph/base.hpp"
template <typename T>
struct Edge {
int frm, to;
T cost;
int id;
};
template <typename T = int, bool directed = false>
struct Graph {
static constexpr bool is_directed = directed;
int N, M;
using cost_type = T;
using edge_type = Edge<T>;
vector<edge_type> edges;
vector<int> indptr;
vector<edge_type> csr_edges;
mutable vc<int> vc_deg, vc_indeg, vc_outdeg;
mutable HashMap<int> MP_FOR_EID;
bool prepared;
class OutgoingEdges {
public:
OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}
const edge_type* begin() const {
if (l == r) {
return 0;
}
return &G->csr_edges[l];
}
const edge_type* end() const {
if (l == r) {
return 0;
}
return &G->csr_edges[r];
}
private:
const Graph* G;
int l, r;
};
bool is_prepared() const { return prepared; }
Graph() : N(0), M(0), prepared(0) {}
Graph(int N) : N(N), M(0), prepared(0) {}
void build(int n) {
N = n, M = 0;
prepared = 0;
edges.clear();
indptr.clear();
csr_edges.clear();
vc_deg.clear();
vc_indeg.clear();
vc_outdeg.clear();
MP_FOR_EID.clear();
}
void add(int frm, int to, T cost = 1, int i = -1) {
assert(!prepared);
assert(0 <= frm && frm < N && 0 <= to && to < N);
if (i == -1) i = M;
auto e = edge_type({frm, to, cost, i});
edges.eb(e);
++M;
}
#ifdef FASTIO
// wt, off
void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }
void read_graph(int M, bool wt = false, int off = 1) {
for (int m = 0; m < M; ++m) {
INT(a, b);
a -= off, b -= off;
if (!wt) {
add(a, b);
} else {
T c;
read(c);
add(a, b, c);
}
}
build();
}
#endif
void build() {
assert(!prepared);
prepared = true;
indptr.assign(N + 1, 0);
for (auto&& e : edges) {
indptr[e.frm + 1]++;
if (!directed) indptr[e.to + 1]++;
}
for (int v = 0; v < N; ++v) {
indptr[v + 1] += indptr[v];
}
auto counter = indptr;
csr_edges.resize(indptr.back() + 1);
for (auto&& e : edges) {
csr_edges[counter[e.frm]++] = e;
if (!directed)
csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
}
}
OutgoingEdges operator[](int v) const {
assert(prepared);
return {this, indptr[v], indptr[v + 1]};
}
vc<int> deg_array() const {
if (vc_deg.empty()) calc_deg();
return vc_deg;
}
pair<vc<int>, vc<int>> deg_array_inout() const {
if (vc_indeg.empty()) calc_deg_inout();
return {vc_indeg, vc_outdeg};
}
int deg(int v) const {
if (vc_deg.empty()) calc_deg();
return vc_deg[v];
}
int in_deg(int v) const {
if (vc_indeg.empty()) calc_deg_inout();
return vc_indeg[v];
}
int out_deg(int v) const {
if (vc_outdeg.empty()) calc_deg_inout();
return vc_outdeg[v];
}
#ifdef FASTIO
void debug() {
#ifdef LOCAL
print("Graph");
if (!prepared) {
print("frm to cost id");
for (auto&& e : edges) print(e.frm, e.to, e.cost, e.id);
} else {
print("indptr", indptr);
print("frm to cost id");
FOR(v, N) for (auto&& e : (*this)[v]) print(e.frm, e.to, e.cost, e.id);
}
flush();
#endif
}
#endif
vc<int> new_idx;
vc<bool> used_e;
// G における頂点 V[i] が、新しいグラフで i になるようにする
// {G, es}
// sum(deg(v)) の計算量になっていて、
// 新しいグラフの n+m より大きい可能性があるので注意
Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
if (len(new_idx) != N) new_idx.assign(N, -1);
int n = len(V);
FOR(i, n) new_idx[V[i]] = i;
Graph<T, directed> G(n);
vc<int> history;
FOR(i, n) {
for (auto&& e : (*this)[V[i]]) {
if (len(used_e) <= e.id) used_e.resize(e.id + 1);
if (used_e[e.id]) continue;
int a = e.frm, b = e.to;
if (new_idx[a] != -1 && new_idx[b] != -1) {
history.eb(e.id);
used_e[e.id] = 1;
int eid = (keep_eid ? e.id : -1);
G.add(new_idx[a], new_idx[b], e.cost, eid);
}
}
}
FOR(i, n) new_idx[V[i]] = -1;
for (auto&& eid : history) used_e[eid] = 0;
G.build();
return G;
}
Graph<T, true> to_directed_tree(int root = -1) const {
if (root == -1) root = 0;
assert(!is_directed && prepared && M == N - 1);
Graph<T, true> G1(N);
vc<int> par(N, -1);
auto dfs = [&](auto& dfs, int v) -> void {
for (auto& e : (*this)[v]) {
if (e.to == par[v]) continue;
par[e.to] = v, dfs(dfs, e.to);
}
};
dfs(dfs, root);
for (auto& e : edges) {
int a = e.frm, b = e.to;
if (par[a] == b) swap(a, b);
assert(par[b] == a);
G1.add(a, b, e.cost);
}
G1.build();
return G1;
}
int get_eid(u64 a, u64 b) const {
if (len(MP_FOR_EID) == 0) {
MP_FOR_EID.build(N - 1);
for (auto& e : edges) {
u64 a = e.frm, b = e.to;
u64 k = to_eid_key(a, b);
MP_FOR_EID[k] = e.id;
}
}
return MP_FOR_EID.get(to_eid_key(a, b), -1);
}
u64 to_eid_key(u64 a, u64 b) const {
if (!directed && a > b) swap(a, b);
return N * a + b;
}
private:
void calc_deg() const {
assert(vc_deg.empty());
vc_deg.resize(N);
for (auto&& e : edges) vc_deg[e.frm]++, vc_deg[e.to]++;
}
void calc_deg_inout() const {
assert(vc_indeg.empty());
vc_indeg.resize(N);
vc_outdeg.resize(N);
for (auto&& e : edges) {
vc_indeg[e.to]++, vc_outdeg[e.frm]++;
}
}
};
#line 3 "graph/tree.hpp"
// HLD euler tour をとっていろいろ
// HLD=false: 入力辺順で preorder
template <typename GT, bool HLD = true>
struct Tree {
using Graph_type = GT;
using WT = typename GT::cost_type;
int N = 0;
vector<int> LID, RID, head, V, parent, VtoE, EtoV;
vc<int> depth;
vc<WT> depth_weighted;
vc<int> memo_tail;
Tree() = default;
Tree(const GT &G, int r = 0) { build(G, r); }
void build(const GT &G, int r = 0) {
N = G.N;
assert(G.is_prepared());
assert(G.M == N - 1);
FOR(i, N - 1) assert(G.edges[i].id == i);
assert(0 <= r && r < N);
if constexpr (!HLD)
build_simple(G, r);
else
build_HLD(G, r);
}
vc<int> heavy_path_at(int v) const {
static_assert(HLD);
assert(head[v] == v);
int k = LID[v];
vc<int> P;
while (k < N && head[V[k]] == v) P.eb(V[k++]);
return P;
}
int heavy_child(int v) const {
static_assert(HLD);
if (RID[v] == LID[v] + 1) return -1;
return V[LID[v] + 1];
}
int tail(int v) {
static_assert(HLD);
if (memo_tail.empty()) {
memo_tail.assign(N, -1);
FOR_R(i, N) {
int v = V[i];
int w = heavy_child(v);
memo_tail[v] = (w == -1 ? v : memo_tail[w]);
}
}
return memo_tail[v];
}
int e_to_v(int eid) const { return EtoV[eid]; }
int v_to_e(int v) const { return VtoE[v]; }
int get_eid(int u, int v) const {
if (parent[u] != v) swap(u, v);
assert(parent[u] == v);
return VtoE[u];
}
int ELID(int v) const { return 2 * LID[v] - depth[v]; }
int ERID(int v) const { return 2 * RID[v] - depth[v] - 1; }
// 目標地点へ進む個数が k
int LA(int v, int k) const {
static_assert(HLD);
assert(k <= depth[v]);
while (1) {
int u = head[v];
if (LID[v] - k >= LID[u]) return V[LID[v] - k];
k -= LID[v] - LID[u] + 1;
v = parent[u];
}
}
int LCA(int u, int v) const {
static_assert(HLD);
for (;; v = parent[head[v]]) {
if (LID[u] > LID[v]) swap(u, v);
if (head[u] == head[v]) return u;
}
}
int meet(int a, int b, int c) const {
static_assert(HLD);
return LCA(a, b) ^ LCA(a, c) ^ LCA(b, c);
}
int subtree_size(int v) const { return RID[v] - LID[v]; }
int subtree_size(int v, int root) const {
static_assert(HLD);
if (v == root) return N;
int x = jump(v, root, 1);
if (in_subtree(v, x)) return RID[v] - LID[v];
return N - RID[x] + LID[x];
}
int dist(int a, int b) const {
static_assert(HLD);
int c = LCA(a, b);
return depth[a] + depth[b] - 2 * depth[c];
}
WT dist_weighted(int a, int b) const {
static_assert(HLD);
int c = LCA(a, b);
return depth_weighted[a] + depth_weighted[b] - WT(2) * depth_weighted[c];
}
// a is in b
bool in_subtree(int a, int b) const {
return LID[b] <= LID[a] && LID[a] < RID[b];
}
int jump(int a, int b, ll k) const {
static_assert(HLD);
if (k == 1) {
if (a == b) return -1;
return (in_subtree(b, a) ? LA(b, depth[b] - depth[a] - 1) : parent[a]);
}
int c = LCA(a, b);
int d_ac = depth[a] - depth[c];
int d_bc = depth[b] - depth[c];
if (k > d_ac + d_bc) return -1;
if (k <= d_ac) return LA(a, k);
return LA(b, d_ac + d_bc - k);
}
vc<int> collect_child(int v) const {
vc<int> res;
for (int k = LID[v] + 1; k < RID[v];) {
res.eb(V[k]), k = RID[V[k]];
}
return res;
}
vc<int> collect_subtree(int v) const {
return {V.begin() + LID[v], V.begin() + RID[v]};
}
vc<int> collect_light(int v) const {
static_assert(HLD);
vc<int> res;
for (int k = LID[v] + 1; k < RID[v];) {
if (head[V[k]] == V[k]) res.eb(V[k]);
k = RID[V[k]];
}
return res;
}
vc<pair<int, int>> get_path_decomposition(int u, int v, bool edge) const {
static_assert(HLD);
// [始点, 終点] の"閉"区間列。
vc<pair<int, int>> up, down;
while (1) {
if (head[u] == head[v]) break;
if (LID[u] < LID[v]) {
down.eb(LID[head[v]], LID[v]);
v = parent[head[v]];
} else {
up.eb(LID[u], LID[head[u]]);
u = parent[head[u]];
}
}
if (LID[u] < LID[v]) down.eb(LID[u] + edge, LID[v]);
elif (LID[v] + edge <= LID[u]) up.eb(LID[u], LID[v] + edge);
reverse(all(down));
up.insert(up.end(), all(down));
return up;
}
// 辺の列の情報 (frm,to,str)
// str = "heavy_up", "heavy_down", "light_up", "light_down"
vc<tuple<int, int, string>> get_path_decomposition_detail(
int u, int v) const {
static_assert(HLD);
vc<tuple<int, int, string>> up, down;
while (1) {
if (head[u] == head[v]) break;
if (LID[u] < LID[v]) {
if (v != head[v]) down.eb(head[v], v, "heavy_down"), v = head[v];
down.eb(parent[v], v, "light_down"), v = parent[v];
} else {
if (u != head[u]) up.eb(u, head[u], "heavy_up"), u = head[u];
up.eb(u, parent[u], "light_up"), u = parent[u];
}
}
if (LID[u] < LID[v]) down.eb(u, v, "heavy_down");
elif (LID[v] < LID[u]) up.eb(u, v, "heavy_up");
reverse(all(down));
concat(up, down);
return up;
}
vc<int> restore_path(int u, int v) const {
vc<int> L, R;
while (depth[u] > depth[v]) L.eb(u), u = parent[u];
while (depth[u] < depth[v]) R.eb(v), v = parent[v];
while (u != v) L.eb(u), R.eb(v), u = parent[u], v = parent[v];
L.eb(u);
while (len(R)) L.eb(POP(R));
return L;
}
// path [a,b] と [c,d] の交わり. 空ならば {-1,-1}.
// https://codeforces.com/problemset/problem/500/G
pair<int, int> path_intersection(int a, int b, int c, int d) const {
static_assert(HLD);
int ab = LCA(a, b), ac = LCA(a, c), ad = LCA(a, d);
int bc = LCA(b, c), bd = LCA(b, d), cd = LCA(c, d);
int x = ab ^ ac ^ bc, y = ab ^ ad ^ bd; // meet(a,b,c), meet(a,b,d)
if (x != y) return {x, y};
int z = ac ^ ad ^ cd;
if (x != z) x = -1;
return {x, x};
}
// uv path 上で check(v) を満たす最後の v
// なければ （つまり check(v) が ng ）-1
template <class F>
int max_path(F check, int u, int v) const {
static_assert(HLD);
if (!check(u)) return -1;
auto pd = get_path_decomposition(u, v, false);
for (auto [a, b] : pd) {
if (!check(V[a])) return u;
if (check(V[b])) {
u = V[b];
continue;
}
int c =
binary_search([&](int c) -> bool { return check(V[c]); }, a, b, 0);
return V[c];
}
return u;
}
private:
void build_simple(const GT &G, int r = 0) {
N = G.N;
LID.assign(N, 0), RID.assign(N, 0);
V.assign(N, -1), parent.assign(N, -1), VtoE.assign(N, -1),
EtoV.assign(N - 1, -1);
depth.assign(N, 0), depth_weighted.assign(N, 0);
// 1st dfs.
int k = 0;
vc<int> st;
st.reserve(N);
st.eb(r);
while (len(st)) {
int v = POP(st);
LID[v] = k, V[k] = v;
++k;
for (int i = G.indptr[v + 1] - 1; i >= G.indptr[v]; --i) {
auto &e = G.csr_edges[i];
if (e.to == parent[v]) continue;
parent[e.to] = v;
depth[e.to] = depth[v] + 1;
depth_weighted[e.to] = depth_weighted[v] + e.cost;
VtoE[e.to] = e.id, EtoV[e.id] = e.to;
st.eb(e.to);
}
}
FOR_R(i, N) {
int v = V[i];
chmax(RID[v], LID[v] + 1);
if (parent[v] != -1) chmax(RID[parent[v]], RID[v]);
}
}
void build_HLD(const GT &G, int r = 0) {
N = G.N;
LID.assign(N, 0), RID.assign(N, 0), head.assign(N, r);
V.assign(N, -1), parent.assign(N, -1), VtoE.assign(N, -1),
EtoV.assign(N - 1, -1);
depth.assign(N, 0), depth_weighted.assign(N, 0);
memo_tail.clear();
// 1st dfs.
{
int k = 0;
vc<int> st;
st.reserve(N);
st.eb(r);
while (len(st)) {
int v = POP(st);
V[k++] = v;
for (auto &e : G[v]) {
if (e.to == parent[v]) continue;
parent[e.to] = v, st.eb(e.to), depth[e.to] = depth[v] + 1;
depth_weighted[e.to] = depth_weighted[v] + e.cost;
VtoE[e.to] = e.id, EtoV[e.id] = e.to;
}
}
// 一時的に RID[v] := sz[v]
FOR_R(i, N) {
int v = V[i];
RID[v] += 1;
if (parent[v] != -1) RID[parent[v]] += RID[v];
}
}
// 2nd dfs.
{
int k = 0;
vc<int> st;
st.reserve(N);
st.eb(r);
while (len(st)) {
int v = POP(st);
V[k] = v, LID[v] = k;
RID[v] = k + RID[v];
++k;
int max_sz = 0, max_ch = -1;
for (auto &e : G[v]) {
if (e.to == parent[v]) continue;
if (chmax(max_sz, RID[e.to])) max_ch = e.to;
}
for (int i = G.indptr[v + 1] - 1; i >= G.indptr[v]; --i) {
auto &e = G.csr_edges[i];
if (e.to == parent[v] || e.to == max_ch) continue;
st.eb(e.to), head[e.to] = e.to;
}
if (max_ch != -1) st.eb(max_ch), head[max_ch] = head[v];
}
}
}
};
#line 4 "graph/ds/tree_wavelet_matrix.hpp"
// https://atcoder.jp/contests/pakencamp-2022-day1/tasks/pakencamp_2022_day1_j
// https://atcoder.jp/contests/utpc2011/tasks/utpc2011_12
template <typename TREE, bool edge, typename T, bool COMPRESS,
typename Monoid = Monoid_Add<T>>
struct Tree_Wavelet_Matrix {
TREE& tree;
int N;
using WM = Wavelet_Matrix<T, COMPRESS, Monoid_Add<T>>;
using X = typename Monoid::value_type;
WM wm;
Tree_Wavelet_Matrix(TREE& tree, vc<T> A, vc<X> SUM_data = {}, int log = -1)
: tree(tree), N(tree.N) {
vc<X>& S = SUM_data;
vc<T> A1;
vc<X> S1;
A1.resize(N);
if (!S.empty()) S1.resize(N);
if (!edge) {
assert(len(A) == N && (len(S) == 0 || len(S) == N));
FOR(v, N) A1[tree.LID[v]] = A[v];
if (len(S) == N) { FOR(v, N) S1[tree.LID[v]] = S[v]; }
wm.build(A1, S1, log);
} else {
assert(len(A) == N - 1 && (len(S) == 0 || len(S) == N - 1));
if (!S.empty()) {
FOR(e, N - 1) { S1[tree.LID[tree.e_to_v(e)]] = S[e]; }
}
FOR(e, N - 1) { A1[tree.LID[tree.e_to_v(e)]] = A[e]; }
wm.build(A1, S1, log);
}
}
// xor した結果で [a, b) に収まるものを数える
int count_path(int s, int t, T a, T b, T xor_val = 0) {
return wm.count(get_segments(s, t), a, b, xor_val);
}
// xor した結果で [a, b) に収まるものを数える
int count_subtree(int u, T a, T b, T xor_val = 0) {
int l = tree.LID[u], r = tree.RID[u];
return wm.count(l + edge, r, a, b, xor_val);
}
// xor した結果で、[L, R) の中で k>=0 番目と prefix sum
pair<T, X> kth_value_and_sum_path(int s, int t, int k, T xor_val = 0) {
return wm.kth_value_and_sum(get_segments(s, t), k, xor_val);
}
// xor した結果で、[L, R) の中で k>=0 番目と prefix sum
pair<T, X> kth_value_and_sum_subtree(int u, int k, T xor_val = 0) {
int l = tree.LID[u], r = tree.RID[u];
return wm.kth_value_and_sum(l + edge, r, k, xor_val);
}
// xor した結果で、[L, R) の中で k>=0 番目
T kth_path(int s, int t, int k, T xor_val = 0) {
return wm.kth(get_segments(s, t), k, xor_val);
}
// xor した結果で、[L, R) の中で k>=0 番目
T kth_subtree(int u, int k, T xor_val = 0) {
int l = tree.LID[u], r = tree.RID[u];
return wm.kth(l + edge, r, k, xor_val);
}
// xor した結果で、[L, R) の中で中央値。
// LOWER = true：下側中央値、false：上側中央値
T median_path(bool UPPER, int s, int t, T xor_val = 0) {
return wm.median(UPPER, get_segments(s, t), xor_val);
}
T median_subtree(bool UPPER, int u, T xor_val = 0) {
int l = tree.LID[u], r = tree.RID[u];
return wm.median(UPPER, l + edge, r, xor_val);
}
// xor した結果で [k1, k2) 番目であるところの SUM_data の和
X sum_path(int s, int t, int k1, int k2, T xor_val = 0) {
return wm.sum(get_segments(s, t), k1, k2, xor_val);
}
// xor した結果で [k1, k2) 番目であるところの SUM_data の和
X sum_subtree(int u, int k1, int k2, T xor_val = 0) {
int l = tree.LID[u], r = tree.RID[u];
return wm.sum(l + edge, r, k1, k2, xor_val);
}
X sum_all_path(int s, int t) { return wm.sum_all(get_segments(s, t)); }
X sum_all_subtree(int u) {
int l = tree.LID[u], r = tree.RID[u];
return wm.sum_all(l + edge, r);
}
private:
vc<pair<int, int>> get_segments(int s, int t) {
vc<pair<int, int>> segments = tree.get_path_decomposition(s, t, edge);
for (auto&& [a, b]: segments) {
if (a >= b) swap(a, b);
++b;
}
return segments;
}
};