cplib-cpp

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:heavy_check_mark: tree/test/frequency_table_of_tree_distance_ntt.test.cpp

Depends on

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"

// Centroid Decomposition
// Verification: https://yukicoder.me/problems/no/2892
// find_current_centroids(int r, int conn_size): Enumerate centroid(s) of the subtree which `r` belongs to.
struct CentroidDecomposition {
    int V;
    std::vector<std::vector<int>> to;

private:
    std::vector<int> is_alive;
    std::vector<int> subtree_size;

    template <class F> void decompose(int r, int conn_size, F callback) {

        const int c = find_current_centroids(r, conn_size).first;
        is_alive.at(c) = 0;

        callback(c);

        for (int nxt : to.at(c)) {
            if (!is_alive.at(nxt)) continue;
            int next_size = subtree_size.at(nxt);
            if (subtree_size.at(nxt) > subtree_size.at(c))
                next_size = subtree_size.at(r) - subtree_size.at(c);
            decompose(nxt, next_size, callback);
        }
    }

public:
    CentroidDecomposition(int v = 0) : V(v), to(v), is_alive(v, 1), subtree_size(v) {}

    CentroidDecomposition(int v, const std::vector<std::pair<int, int>> &tree_edges)
        : CentroidDecomposition(v) {
        for (auto e : tree_edges) add_edge(e.first, e.second);
    }

    void add_edge(int v1, int v2) {
        assert(0 <= v1 and v1 < V and 0 <= v2 and v2 < V);
        assert(v1 != v2);
        to.at(v1).push_back(v2), to.at(v2).emplace_back(v1);
    }

    std::pair<int, int> find_current_centroids(int r, int conn_size) {
        assert(is_alive.at(r));

        const int thres = conn_size / 2;

        int c1 = -1, c2 = -1;

        auto rec_search = [&](auto &&self, int now, int prv) -> void {
            bool is_centroid = true;
            subtree_size.at(now) = 1;
            for (int nxt : to.at(now)) {
                if (nxt == prv or !is_alive.at(nxt)) continue;
                self(self, nxt, now);
                subtree_size.at(now) += subtree_size.at(nxt);
                if (subtree_size.at(nxt) > thres) is_centroid = false;
            }
            if (conn_size - subtree_size.at(now) > thres) is_centroid = false;

            if (is_centroid) (c1 < 0 ? c1 : c2) = now;
        };
        rec_search(rec_search, r, -1);

        return {c1, c2};
    }

    template <class F> void run(int r, F callback) {
        int conn_size = 0;

        auto rec = [&](auto &&self, int now, int prv) -> void {
            ++conn_size;
            is_alive.at(now) = 1;

            for (int nxt : to.at(now)) {
                if (nxt == prv) continue;
                self(self, nxt, now);
            }
        };
        rec(rec, r, -1);

        decompose(r, conn_size, callback);
    }

    std::vector<int> centroid_decomposition(int r) {
        std::vector<int> res;
        run(r, [&](int v) { res.push_back(v); });
        return res;
    }
};
#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;

        CentroidDecomposition c(to.size());
        for (int i = 0; i < int(to.size()); i++) {
            for (int j : to[i]) {
                if (i < j) c.add_edge(i, j);
            }
        }

        cd = c.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';
}
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