cplib-cpp
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tree/test/frequency_table_of_tree_distance_ntt.test.cpp
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- Last update: 2023-12-26 21:26:22+09:00
- Problem: https://judge.yosupo.jp/problem/frequency_table_of_tree_distance
Depends on
- convolution/ntt.hpp
- modint.hpp
- number/bare_mod_algebra.hpp
- tree/centroid_decomposition.hpp
- Frequency table of tree distance
(tree/frequency_table_of_tree_distance.hpp)
Code
#include "../../convolution/ntt.hpp"
#include "../../modint.hpp"
#include "../../number/bare_mod_algebra.hpp"
#include "../frequency_table_of_tree_distance.hpp"
#include <iostream>
#include <vector>
#define PROBLEM "https://judge.yosupo.jp/problem/frequency_table_of_tree_distance"
using namespace std;
int main() {
cin.tie(nullptr), ios::sync_with_stdio(false);
int N;
cin >> N;
vector<vector<int>> to(N);
for (int i = 0; i < N - 1; i++) {
int s, t;
cin >> s >> t;
to[s].emplace_back(t), to[t].emplace_back(s);
}
frequency_table_of_tree_distance solver(to);
using mint1 = ModInt<998244353>;
using mint2 = ModInt<167772161>;
const vector<mint1> ret1 =
frequency_table_of_tree_distance(to).solve<mint1, nttconv<mint1>>(std::vector<mint1>(N, 1));
const vector<mint2> ret2 =
frequency_table_of_tree_distance(to).solve<mint2, nttconv<mint2>>(std::vector<mint2>(N, 1));
for (int i = 1; i < N; i++) {
auto v = crt<long long>({ret1[i].val(), ret2[i].val()}, {mint1::mod(), mint2::mod()});
cout << v.first << ' ';
}
cout << '\n';
}#line 2 "modint.hpp"
#include <cassert>
#include <iostream>
#include <set>
#include <vector>
template <int md> struct ModInt {
using lint = long long;
constexpr static int mod() { return md; }
static int get_primitive_root() {
static int primitive_root = 0;
if (!primitive_root) {
primitive_root = [&]() {
std::set<int> fac;
int v = md - 1;
for (lint i = 2; i * i <= v; i++)
while (v % i == 0) fac.insert(i), v /= i;
if (v > 1) fac.insert(v);
for (int g = 1; g < md; g++) {
bool ok = true;
for (auto i : fac)
if (ModInt(g).pow((md - 1) / i) == 1) {
ok = false;
break;
}
if (ok) return g;
}
return -1;
}();
}
return primitive_root;
}
int val_;
int val() const noexcept { return val_; }
constexpr ModInt() : val_(0) {}
constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; }
constexpr ModInt(lint v) { _setval(v % md + md); }
constexpr explicit operator bool() const { return val_ != 0; }
constexpr ModInt operator+(const ModInt &x) const {
return ModInt()._setval((lint)val_ + x.val_);
}
constexpr ModInt operator-(const ModInt &x) const {
return ModInt()._setval((lint)val_ - x.val_ + md);
}
constexpr ModInt operator*(const ModInt &x) const {
return ModInt()._setval((lint)val_ * x.val_ % md);
}
constexpr ModInt operator/(const ModInt &x) const {
return ModInt()._setval((lint)val_ * x.inv().val() % md);
}
constexpr ModInt operator-() const { return ModInt()._setval(md - val_); }
constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; }
friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; }
friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; }
friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; }
constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; }
constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; }
constexpr bool operator<(const ModInt &x) const {
return val_ < x.val_;
} // To use std::map<ModInt, T>
friend std::istream &operator>>(std::istream &is, ModInt &x) {
lint t;
return is >> t, x = ModInt(t), is;
}
constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {
return os << x.val_;
}
constexpr ModInt pow(lint n) const {
ModInt ans = 1, tmp = *this;
while (n) {
if (n & 1) ans *= tmp;
tmp *= tmp, n >>= 1;
}
return ans;
}
static constexpr int cache_limit = std::min(md, 1 << 21);
static std::vector<ModInt> facs, facinvs, invs;
constexpr static void _precalculation(int N) {
const int l0 = facs.size();
if (N > md) N = md;
if (N <= l0) return;
facs.resize(N), facinvs.resize(N), invs.resize(N);
for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
facinvs[N - 1] = facs.back().pow(md - 2);
for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
}
constexpr ModInt inv() const {
if (this->val_ < cache_limit) {
if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0};
while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
return invs[this->val_];
} else {
return this->pow(md - 2);
}
}
constexpr ModInt fac() const {
while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
return facs[this->val_];
}
constexpr ModInt facinv() const {
while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
return facinvs[this->val_];
}
constexpr ModInt doublefac() const {
lint k = (this->val_ + 1) / 2;
return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac())
: ModInt(k).fac() * ModInt(2).pow(k);
}
constexpr ModInt nCr(int r) const {
if (r < 0 or this->val_ < r) return ModInt(0);
return this->fac() * (*this - r).facinv() * ModInt(r).facinv();
}
constexpr ModInt nPr(int r) const {
if (r < 0 or this->val_ < r) return ModInt(0);
return this->fac() * (*this - r).facinv();
}
static ModInt binom(int n, int r) {
static long long bruteforce_times = 0;
if (r < 0 or n < r) return ModInt(0);
if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r);
r = std::min(r, n - r);
ModInt ret = ModInt(r).facinv();
for (int i = 0; i < r; ++i) ret *= n - i;
bruteforce_times += r;
return ret;
}
// Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!)
// Complexity: O(sum(ks))
template <class Vec> static ModInt multinomial(const Vec &ks) {
ModInt ret{1};
int sum = 0;
for (int k : ks) {
assert(k >= 0);
ret *= ModInt(k).facinv(), sum += k;
}
return ret * ModInt(sum).fac();
}
// Catalan number, C_n = binom(2n, n) / (n + 1)
// C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ...
// https://oeis.org/A000108
// Complexity: O(n)
static ModInt catalan(int n) {
if (n < 0) return ModInt(0);
return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv();
}
ModInt sqrt() const {
if (val_ == 0) return 0;
if (md == 2) return val_;
if (pow((md - 1) / 2) != 1) return 0;
ModInt b = 1;
while (b.pow((md - 1) / 2) == 1) b += 1;
int e = 0, m = md - 1;
while (m % 2 == 0) m >>= 1, e++;
ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
x *= (*this);
ModInt z = b.pow(m);
while (y != 1) {
int j = 0;
ModInt t = y;
while (t != 1) j++, t *= t;
z = z.pow(1LL << (e - j - 1));
x *= z, z *= z, y *= z;
e = j;
}
return ModInt(std::min(x.val_, md - x.val_));
}
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};
using ModInt998244353 = ModInt<998244353>;
// using mint = ModInt<998244353>;
// using mint = ModInt<1000000007>;
#line 3 "convolution/ntt.hpp"
#include <algorithm>
#include <array>
#line 7 "convolution/ntt.hpp"
#include <tuple>
#line 9 "convolution/ntt.hpp"
// CUT begin
// Integer convolution for arbitrary mod
// with NTT (and Garner's algorithm) for ModInt / ModIntRuntime class.
// We skip Garner's algorithm if `skip_garner` is true or mod is in `nttprimes`.
// input: a (size: n), b (size: m)
// return: vector (size: n + m - 1)
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner);
constexpr int nttprimes[3] = {998244353, 167772161, 469762049};
// Integer FFT (Fast Fourier Transform) for ModInt class
// (Also known as Number Theoretic Transform, NTT)
// is_inverse: inverse transform
// ** Input size must be 2^n **
template <typename MODINT> void ntt(std::vector<MODINT> &a, bool is_inverse = false) {
int n = a.size();
if (n == 1) return;
static const int mod = MODINT::mod();
static const MODINT root = MODINT::get_primitive_root();
assert(__builtin_popcount(n) == 1 and (mod - 1) % n == 0);
static std::vector<MODINT> w{1}, iw{1};
for (int m = w.size(); m < n / 2; m *= 2) {
MODINT dw = root.pow((mod - 1) / (4 * m)), dwinv = 1 / dw;
w.resize(m * 2), iw.resize(m * 2);
for (int i = 0; i < m; i++) w[m + i] = w[i] * dw, iw[m + i] = iw[i] * dwinv;
}
if (!is_inverse) {
for (int m = n; m >>= 1;) {
for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
for (int i = s; i < s + m; i++) {
MODINT x = a[i], y = a[i + m] * w[k];
a[i] = x + y, a[i + m] = x - y;
}
}
}
} else {
for (int m = 1; m < n; m *= 2) {
for (int s = 0, k = 0; s < n; s += 2 * m, k++) {
for (int i = s; i < s + m; i++) {
MODINT x = a[i], y = a[i + m];
a[i] = x + y, a[i + m] = (x - y) * iw[k];
}
}
}
int n_inv = MODINT(n).inv().val();
for (auto &v : a) v *= n_inv;
}
}
template <int MOD>
std::vector<ModInt<MOD>> nttconv_(const std::vector<int> &a, const std::vector<int> &b) {
int sz = a.size();
assert(a.size() == b.size() and __builtin_popcount(sz) == 1);
std::vector<ModInt<MOD>> ap(sz), bp(sz);
for (int i = 0; i < sz; i++) ap[i] = a[i], bp[i] = b[i];
ntt(ap, false);
if (a == b)
bp = ap;
else
ntt(bp, false);
for (int i = 0; i < sz; i++) ap[i] *= bp[i];
ntt(ap, true);
return ap;
}
long long garner_ntt_(int r0, int r1, int r2, int mod) {
using mint2 = ModInt<nttprimes[2]>;
static const long long m01 = 1LL * nttprimes[0] * nttprimes[1];
static const long long m0_inv_m1 = ModInt<nttprimes[1]>(nttprimes[0]).inv().val();
static const long long m01_inv_m2 = mint2(m01).inv().val();
int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1];
auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * m01_inv_m2;
return (r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val()) % mod;
}
template <typename MODINT>
std::vector<MODINT> nttconv(std::vector<MODINT> a, std::vector<MODINT> b, bool skip_garner) {
if (a.empty() or b.empty()) return {};
int sz = 1, n = a.size(), m = b.size();
while (sz < n + m) sz <<= 1;
if (sz <= 16) {
std::vector<MODINT> ret(n + m - 1);
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) ret[i + j] += a[i] * b[j];
}
return ret;
}
int mod = MODINT::mod();
if (skip_garner or
std::find(std::begin(nttprimes), std::end(nttprimes), mod) != std::end(nttprimes)) {
a.resize(sz), b.resize(sz);
if (a == b) {
ntt(a, false);
b = a;
} else {
ntt(a, false), ntt(b, false);
}
for (int i = 0; i < sz; i++) a[i] *= b[i];
ntt(a, true);
a.resize(n + m - 1);
} else {
std::vector<int> ai(sz), bi(sz);
for (int i = 0; i < n; i++) ai[i] = a[i].val();
for (int i = 0; i < m; i++) bi[i] = b[i].val();
auto ntt0 = nttconv_<nttprimes[0]>(ai, bi);
auto ntt1 = nttconv_<nttprimes[1]>(ai, bi);
auto ntt2 = nttconv_<nttprimes[2]>(ai, bi);
a.resize(n + m - 1);
for (int i = 0; i < n + m - 1; i++)
a[i] = garner_ntt_(ntt0[i].val(), ntt1[i].val(), ntt2[i].val(), mod);
}
return a;
}
template <typename MODINT>
std::vector<MODINT> nttconv(const std::vector<MODINT> &a, const std::vector<MODINT> &b) {
return nttconv<MODINT>(a, b, false);
}
#line 5 "number/bare_mod_algebra.hpp"
#include <utility>
#line 7 "number/bare_mod_algebra.hpp"
// CUT begin
// Solve ax+by=gcd(a, b)
template <class Int> Int extgcd(Int a, Int b, Int &x, Int &y) {
Int d = a;
if (b != 0) {
d = extgcd(b, a % b, y, x), y -= (a / b) * x;
} else {
x = 1, y = 0;
}
return d;
}
// Calculate a^(-1) (MOD m) s if gcd(a, m) == 1
// Calculate x s.t. ax == gcd(a, m) MOD m
template <class Int> Int mod_inverse(Int a, Int m) {
Int x, y;
extgcd<Int>(a, m, x, y);
x %= m;
return x + (x < 0) * m;
}
// Require: 1 <= b
// return: (g, x) s.t. g = gcd(a, b), xa = g MOD b, 0 <= x < b/g
template <class Int> /* constexpr */ std::pair<Int, Int> inv_gcd(Int a, Int b) {
a %= b;
if (a < 0) a += b;
if (a == 0) return {b, 0};
Int s = b, t = a, m0 = 0, m1 = 1;
while (t) {
Int u = s / t;
s -= t * u, m0 -= m1 * u;
auto tmp = s;
s = t, t = tmp, tmp = m0, m0 = m1, m1 = tmp;
}
if (m0 < 0) m0 += b / s;
return {s, m0};
}
template <class Int>
/* constexpr */ std::pair<Int, Int> crt(const std::vector<Int> &r, const std::vector<Int> &m) {
assert(r.size() == m.size());
int n = int(r.size());
// Contracts: 0 <= r0 < m0
Int r0 = 0, m0 = 1;
for (int i = 0; i < n; i++) {
assert(1 <= m[i]);
Int r1 = r[i] % m[i], m1 = m[i];
if (r1 < 0) r1 += m1;
if (m0 < m1) {
std::swap(r0, r1);
std::swap(m0, m1);
}
if (m0 % m1 == 0) {
if (r0 % m1 != r1) return {0, 0};
continue;
}
Int g, im;
std::tie(g, im) = inv_gcd<Int>(m0, m1);
Int u1 = m1 / g;
if ((r1 - r0) % g) return {0, 0};
Int x = (r1 - r0) / g % u1 * im % u1;
r0 += x * m0;
m0 *= u1;
if (r0 < 0) r0 += m0;
}
return {r0, m0};
}
// 蟻本 P.262
// 中国剰余定理を利用して，色々な素数で割った余りから元の値を復元
// 連立線形合同式 A * x = B mod M の解
// Requirement: M[i] > 0
// Output: x = first MOD second (if solution exists), (0, 0) (otherwise)
template <class Int>
std::pair<Int, Int>
linear_congruence(const std::vector<Int> &A, const std::vector<Int> &B, const std::vector<Int> &M) {
Int r = 0, m = 1;
assert(A.size() == M.size());
assert(B.size() == M.size());
for (int i = 0; i < (int)A.size(); i++) {
assert(M[i] > 0);
const Int ai = A[i] % M[i];
Int a = ai * m, b = B[i] - ai * r, d = std::__gcd(M[i], a);
if (b % d != 0) {
return std::make_pair(0, 0); // 解なし
}
Int t = b / d * mod_inverse<Int>(a / d, M[i] / d) % (M[i] / d);
r += m * t;
m *= M[i] / d;
}
return std::make_pair((r < 0 ? r + m : r), m);
}
template <class Int = int, class Long = long long> Int pow_mod(Int x, long long n, Int md) {
static_assert(sizeof(Int) * 2 <= sizeof(Long), "Watch out for overflow");
if (md == 1) return 0;
Int ans = 1;
while (n > 0) {
if (n & 1) ans = (Long)ans * x % md;
x = (Long)x * x % md;
n >>= 1;
}
return ans;
}
#line 5 "tree/centroid_decomposition.hpp"
// CUT begin
/*
(Recursive) Centroid Decomposition
Verification: Codeforces #190 Div.1 C https://codeforces.com/contest/321/submission/59093583
fix_root(int r): Build information of the tree which `r` belongs to.
detect_centroid(int r): Enumerate centroid(s) of the tree which `r` belongs to.
*/
struct CentroidDecomposition {
int NO_PARENT = -1;
int V;
int E;
std::vector<std::vector<std::pair<int, int>>> to; // (node_id, edge_id)
std::vector<int> par; // parent node_id par[root] = -1
std::vector<std::vector<int>> chi; // children id's
std::vector<int> subtree_size; // size of each subtree
std::vector<int> available_edge; // If 0, ignore the corresponding edge.
CentroidDecomposition(int v = 0)
: V(v), E(0), to(v), par(v, NO_PARENT), chi(v), subtree_size(v) {}
CentroidDecomposition(const std::vector<std::vector<int>> &to_)
: CentroidDecomposition(to_.size()) {
for (int i = 0; i < V; i++) {
for (auto j : to_[i]) {
if (i < j) { add_edge(i, j); }
}
}
}
void add_edge(int v1, int v2) {
to[v1].emplace_back(v2, E), to[v2].emplace_back(v1, E), E++;
available_edge.emplace_back(1);
}
int _dfs_fixroot(int now, int prv) {
chi[now].clear(), subtree_size[now] = 1;
for (auto nxt : to[now]) {
if (nxt.first != prv and available_edge[nxt.second]) {
par[nxt.first] = now, chi[now].push_back(nxt.first);
subtree_size[now] += _dfs_fixroot(nxt.first, now);
}
}
return subtree_size[now];
}
void fix_root(int root) {
par[root] = NO_PARENT;
_dfs_fixroot(root, -1);
}
//// Centroid Decpmposition ////
std::vector<int> centroid_cand_tmp;
void _dfs_detect_centroids(int now, int prv, int n) {
bool is_centroid = true;
for (auto nxt : to[now]) {
if (nxt.first != prv and available_edge[nxt.second]) {
_dfs_detect_centroids(nxt.first, now, n);
if (subtree_size[nxt.first] > n / 2) is_centroid = false;
}
}
if (n - subtree_size[now] > n / 2) is_centroid = false;
if (is_centroid) centroid_cand_tmp.push_back(now);
}
std::pair<int, int> detect_centroids(int r) { // ([centroid_node_id1], ([centroid_node_id2]|-1))
centroid_cand_tmp.clear();
while (par[r] != NO_PARENT) r = par[r];
int n = subtree_size[r];
_dfs_detect_centroids(r, -1, n);
if (centroid_cand_tmp.size() == 1)
return std::make_pair(centroid_cand_tmp[0], -1);
else
return std::make_pair(centroid_cand_tmp[0], centroid_cand_tmp[1]);
}
std::vector<int> _cd_vertices;
void _centroid_decomposition(int now) {
fix_root(now);
now = detect_centroids(now).first;
_cd_vertices.emplace_back(now);
/*
do something
*/
for (auto p : to[now]) {
int nxt, eid;
std::tie(nxt, eid) = p;
if (available_edge[eid] == 0) continue;
available_edge[eid] = 0;
_centroid_decomposition(nxt);
}
}
std::vector<int> centroid_decomposition(int x) {
_cd_vertices.clear();
_centroid_decomposition(x);
return _cd_vertices;
}
};
#line 5 "tree/frequency_table_of_tree_distance.hpp"
struct frequency_table_of_tree_distance {
std::vector<std::vector<int>> tos;
std::vector<int> cd;
std::vector<std::pair<int, int>> tmp;
std::vector<int> alive;
void _dfs(int now, int prv, int depth) {
// if (int(tmp.size()) <= depth) tmp.resize(depth + 1, 0);
// tmp[depth]++;
tmp.emplace_back(now, depth);
for (auto nxt : tos[now]) {
if (alive[nxt] and nxt != prv) _dfs(nxt, now, depth + 1);
}
}
std::vector<std::pair<int, int>> cnt_dfs(int root) {
return tmp.clear(), _dfs(root, -1, 0), tmp;
}
frequency_table_of_tree_distance(const std::vector<std::vector<int>> &to) {
tos = to;
cd = CentroidDecomposition(to).centroid_decomposition(0);
}
template <class S, std::vector<S> (*conv)(const std::vector<S> &, const std::vector<S> &)>
std::vector<S> solve(const std::vector<S> &weight) {
alive.assign(tos.size(), 1);
std::vector<S> ret(tos.size());
std::vector<S> v;
for (auto root : cd) {
std::vector<std::vector<S>> vv;
alive[root] = 0;
for (auto nxt : tos[root]) {
if (!alive[nxt]) continue;
v.clear();
for (auto p : cnt_dfs(nxt)) {
while (int(v.size()) <= p.second) v.push_back(S(0));
v[p.second] += weight[p.first];
}
for (int i = 0; i < int(v.size()); i++) ret[i + 1] += v[i] * weight[root];
vv.emplace_back(v);
}
std::sort(vv.begin(), vv.end(), [&](const std::vector<S> &l, const std::vector<S> &r) {
return l.size() < r.size();
});
for (size_t j = 1; j < vv.size(); j++) {
const std::vector<S> c = conv(vv[j - 1], vv[j]);
for (size_t i = 0; i < c.size(); i++) ret[i + 2] += c[i];
for (size_t i = 0; i < vv[j - 1].size(); i++) vv[j][i] += vv[j - 1][i];
}
tos[root].clear();
}
return ret;
}
};
#line 7 "tree/test/frequency_table_of_tree_distance_ntt.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/frequency_table_of_tree_distance"
using namespace std;
int main() {
cin.tie(nullptr), ios::sync_with_stdio(false);
int N;
cin >> N;
vector<vector<int>> to(N);
for (int i = 0; i < N - 1; i++) {
int s, t;
cin >> s >> t;
to[s].emplace_back(t), to[t].emplace_back(s);
}
frequency_table_of_tree_distance solver(to);
using mint1 = ModInt<998244353>;
using mint2 = ModInt<167772161>;
const vector<mint1> ret1 =
frequency_table_of_tree_distance(to).solve<mint1, nttconv<mint1>>(std::vector<mint1>(N, 1));
const vector<mint2> ret2 =
frequency_table_of_tree_distance(to).solve<mint2, nttconv<mint2>>(std::vector<mint2>(N, 1));
for (int i = 1; i < N; i++) {
auto v = crt<long long>({ret1[i].val(), ret2[i].val()}, {mint1::mod(), mint2::mod()});
cout << v.first << ' ';
}
cout << '\n';
}