po167_library

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:heavy_check_mark: test/fps/pow.test.cpp

Depends on

Code

#define PROBLEM "https://judge.yosupo.jp/problem/pow_of_formal_power_series"

#include "../../fps/FPS_pow.hpp"

#include <vector>
#include <iostream>
#include <atcoder/modint>


int main(){
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr);
    int N;
    long long M;
    std::cin >> N >> M;
    std::vector<atcoder::modint998244353> A(N);
    for (int i = 0; i < N; i++){
        int a;
        std::cin >> a;
        A[i] = a;
    }
    auto B = po167::FPS_pow(A, M);
    for (int i = 0; i < N; i++){
        if (i) std::cout << " ";
        std::cout << B[i].val();
    }
    std::cout << "\n";
}
#line 1 "test/fps/pow.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/pow_of_formal_power_series"

#line 2 "fps/FPS_pow.hpp"

#include <vector>
#include <atcoder/convolution>

#line 4 "fps/FPS_cyclic_convolution.hpp"

namespace po167{
// |f| = |g| = 2 ^ n
template<class T>
std::vector<T> FPS_cyclic_convolution(std::vector<T> f, std::vector<T> g){
    atcoder::internal::butterfly(f);
    atcoder::internal::butterfly(g);
    for (int i = 0; i < (int)f.size(); i++) f[i] *= g[i];
    atcoder::internal::butterfly_inv(f);
    T iz = (T)(1) / (T)(f.size());
    for (int i = 0; i < (int)f.size(); i++) f[i] *= iz;
    return f;
}
}
#line 3 "fps/FPS_differetial.hpp"

namespace po167{
// return f'
template <class T>
std::vector<T> FPS_differential(std::vector<T> f){
    if (f.empty()) return f;
    for (int i = 0; i < (int)f.size() - 1; i++){
        f[i] = f[i + 1] * (T)(i + 1);
    }
    f.pop_back();
    return f;
}
}
#line 3 "fps/FPS_integral.hpp"

namespace po167{
template <class T>
std::vector<T> FPS_integral(std::vector<T> f){
    if (f.empty()) return f;
    std::vector<T> num_inv((int)f.size() + 1);
    num_inv[0] = 1;
    num_inv[1] = 1;
    auto m = T::mod();
    for (int i = 2; i <= (int)f.size(); i++){
        num_inv[i] = (0 - num_inv[m % i]) * (T)(m / i);
    }
    f.reserve((int)f.size() + 1);
    f.push_back(0);
    for (int i = (int)f.size() - 1; i > 0; i--){
        f[i] = f[i - 1] * num_inv[i];
    }
    f[0] = 0;
    return f;
}
}
#line 4 "fps/FPS_inv.hpp"

namespace po167{
// return 1 / f
template <class T>
std::vector<T> FPS_inv(std::vector<T> f, int len = -1){
    if (len == -1) len = f.size();
    assert(f[0] != 0);
    std::vector<T> g = {1 / f[0]};
    int s = 1;
    while(s < len){
        // g = 2g_s - f(g_s)^2 (mod x ^ (2 * s))
        // g = g - (fg - 1)g
        // (fg - 1) = 0 (mod x ^ (s))
        std::vector<T> n_g(s * 2, 0);
        std::vector<T> f_s(s * 2, 0);
        g.resize(s * 2);
        for (int i = 0; i < s * 2; i++){
            if (int(f.size()) > i) f_s[i] = f[i];
            n_g[i] = g[i];
        }
        atcoder::internal::butterfly(g);
        atcoder::internal::butterfly(f_s);
        for (int i = 0; i < s * 2; i++){
            f_s[i] *= g[i];
        }
        atcoder::internal::butterfly_inv(f_s);
        T iz = 1 / (T)(s * 2);
        for (int i = s; i < s * 2; i++){
            f_s[i] *= iz;
        }
        for (int i = 0; i < s; i++){
            f_s[i] = 0;
        }
        atcoder::internal::butterfly(f_s);
        for (int i = 0; i < s * 2; i++){
            f_s[i] *= g[i];
        }
        atcoder::internal::butterfly_inv(f_s);
        for (int i = s; i < s * 2; i++){
            n_g[i] -= f_s[i] * iz;
        }
        std::swap(n_g, g);
        s *= 2;
    }
    g.resize(len);
    return g;
}
}
#line 8 "fps/FPS_exp.hpp"

namespace po167{
template<class T>
std::vector<T> FPS_exp(std::vector<T> f, int len = -1){
    if (len == -1) len = f.size();
    if (len == 0) return {};
    if (len == 1) return {T(1)};
    assert(!f.empty() && f[0] == 0);
    int s = 1;
    // simple
    std::vector<T> g = {T(1)};
    while (s < len){
        // g' / g
        // A * B
        std::vector<T> A = g, B = g;
        A = FPS_differential(A);
        B = FPS_inv(B, 2 * s);
        A.resize(2 * s);
        A = FPS_cyclic_convolution(A, B);
        A.pop_back();
        A = FPS_integral(A);
        for (int i = 0; i < s; i++) A[i] = 0;
        for (int i = s; i < s * 2; i++) A[i] = (i < (int)f.size() ? f[i] : 0) - A[i];
        // g_hat = g (1 - g + f)
        // g += B = g * A
        g.resize(2 * s);
        B = FPS_cyclic_convolution(A, g);
        for (int i = s; i < s * 2; i++) g[i] = B[i];
        s *= 2;
    }
    g.resize(len);
    return g;
}
}
#line 6 "fps/FPS_log.hpp"

namespace po167{
template<class T>
std::vector<T> FPS_log(std::vector<T> f, int len = -1){
    if (len == -1) len = f.size();
    if (len == 0) return {};
    if (len == 1) return {T(0)};
    assert(!f.empty() && f[0] == 1);
    std::vector<T> res = atcoder::convolution(FPS_differential(f), FPS_inv(f, len));
    res.resize(len - 1);
    return FPS_integral(res);
}
}
#line 8 "fps/FPS_pow.hpp"

namespace po167{
template<class T>
std::vector<T> FPS_pow(std::vector<T> f,long long M, int len = -1){
    if (len == -1) len = f.size();
    std::vector<T> res(len, 0);
    if (M == 0){
        res[0] = 1;
        return res;
    }
    for (int i = 0; i < (int)f.size(); i++){
        if (f[i] == 0) continue;
        if (i > (len - 1) / M) break;
        std::vector<T> g((int)f.size() - i);
        T v = (T)(1) / (T)(f[i]);
        for (int j = i; j < (int)f.size(); j++){
            g[j - i] = f[j] * v;
        }
        long long zero = i * M;
        if (i) len -= i * M;
        g = FPS_log(g, len);
        for (T &x : g) x *= M;
        g = FPS_exp(g, len);
        v = (T)(1) / v;
        T c = 1;
        while (M){
            if (M & 1) c = c * v;
            v = v * v;
            M >>= 1;
        }
        for (int i = 0; i < len; i++) res[i + zero] = g[i] * c;
        return res;
    }
    return res;
}
}
#line 4 "test/fps/pow.test.cpp"

#line 6 "test/fps/pow.test.cpp"
#include <iostream>
#include <atcoder/modint>


int main(){
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr);
    int N;
    long long M;
    std::cin >> N >> M;
    std::vector<atcoder::modint998244353> A(N);
    for (int i = 0; i < N; i++){
        int a;
        std::cin >> a;
        A[i] = a;
    }
    auto B = po167::FPS_pow(A, M);
    for (int i = 0; i < N; i++){
        if (i) std::cout << " ";
        std::cout << B[i].val();
    }
    std::cout << "\n";
}
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