po167_library

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:heavy_check_mark: test/fps/consecutive_linear.test.cpp

Depends on

Code

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


#include "../../fps/FPS_consecutive_linear.hpp"
#include<iostream>
int main(){
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr);
    using mint = atcoder::modint998244353;

    long long d, k, M, num;
    std::cin >> d >> k >> M;
    std::vector<mint> A(d), C(d + 1);
    for (int i = 0; i < d; i++) std::cin >> num, A[i] = num;
    for (int i = 1; i <= d; i++) std::cin >> num, C[i] = num;
    C[0] = -1;
    auto ans = po167::Consecutive_Linear(k, k + M, A, C);
    for (int i = 0; i < (int)ans.size(); i++){
        if (i) std::cout << " ";
        std::cout << ans[i].val();
    }
    std::cout << "\n";
}
#line 1 "test/fps/consecutive_linear.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/consecutive_terms_of_linear_recurrent_sequence"


#line 2 "fps/FPS_consecutive_linear.hpp"
#include<atcoder/convolution>
#line 2 "fps/FPS_inv.hpp"
#include <vector>
#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 3 "fps/FPS_pick_even_odd.hpp"

namespace po167{
// s.t |v| = 2 ^ s (no assert)
template<class T>
void FPS_pick_even_odd(std::vector<T> &v, int odd){
    int z = v.size() / 2;
    T half = (T)(1) / (T)(2);
    if (odd == 0){
        for (int i = 0; i < z; i++){
            v[i] = (v[i * 2] + v[i * 2 + 1]) * half;
        }
        v.resize(z);
    } else {
        T e = (T(atcoder::internal::primitive_root_constexpr(T::mod()))).pow(T::mod() / (2 * z));
        T ie = T(1) / e;
        std::vector<T> es = {half};
        while ((int)es.size() != z){
            std::vector<T> n_es((int)es.size() * 2);
            for (int i = 0; i < (int)es.size(); i++){
                n_es[i * 2] = (es[i]);
                n_es[i * 2 + 1] = (es[i] * ie);
            }
            ie *= ie;
            std::swap(n_es, es);
        }
        for (int i = 0; i < z; i ++){
            v[i] = (v[i * 2] - v[i * 2 + 1]) * es[i];
        }
        v.resize(z);
    }
}
}
#line 5 "fps/FPS_consecutive_linear.hpp"

namespace po167
{
template<class T>
// input A(x)
// B(x) = 1 / A(x)
// return {B[l], B[l + 1], ... , B[r - 1]}
// 0 < r
std::vector<T> FPS_Inv_Consecutive(long long l, long long r, std::vector<T> A){
    assert(0 < r);
    if (l >= r) return {};
    if (r == 1){
        std::vector<T> res(r - l, 0);
        res.back() = (T)(1) / (T)(A[0]);
        return res;
    }
    if (r < (int)A.size() * 4){
        auto res = FPS_inv(A, r);
        std::vector<T> n_res(r - l);
        for (int i = 0; i < r - l; i++){
            if (0 <= i + l && i + l < r) n_res[i] = res[i + l];
            else n_res[i] = 0;
        }
        return n_res;
    }
    int d = A.size();
    int z = 1;
    while ((1 << z) < d) z++;
    std::vector<T> nA(d);
    for (int i = 0; i < d; i++){
        nA[i] = A[i] * (1 - 2 * (i & 1));
    }
    std::vector<T> nC(d);
    // for (int i = 0; i < d; i++) nC[i] = C[i * 2];
    {
        A.resize(1 << (z + 1));
        atcoder::internal::butterfly(A);
        for (int i = 0; i < (1 << z); i++){
            A[i * 2] *= A[i * 2 + 1];
            A[i * 2 + 1] = A[i * 2];
        }
        FPS_pick_even_odd(A, 0);
        atcoder::internal::butterfly_inv(A);
        T iz = ((T)(1)) / ((T)(1 << z));
        for (int i = 0; i < d; i++) nC[i] = A[i] * iz;
    }
    // calc [l - d + 1, r) 1 / C
    // calc [(l - d + 1) / 2, r / 2) 1 / nC
    long long nl = std::max(0ll, (l - d) / 2);
    long long nr = (r + 1) / 2;
    auto D = FPS_Inv_Consecutive(nl, nr, nC);
    std::vector<T> nD(2 * (int)D.size());
    for (int i = 0; i < (int)D.size(); i++){
        nD[i * 2] = D[i];
    }
    auto B = atcoder::convolution(nD, nA);
    std::vector<T> res(r - l);
    for (long long i = l; i < r; i++){
        if (i >= 0) res[i - l] = B[i - nl * 2];
    }
    return res;
}
template<class T>
// 0 = a[i] * c[0] + a[i - 1] * c[1] + a[i - 2] * c[2] + ... + a[i - d] * c[d]
// a.size() + 1 == c.size()
// almost c[0] = - 1 
// return {a[l], a[l + 1], ... , a[r - 1]}
std::vector<T> Consecutive_Linear(long long l, long long r, std::vector<T> a, std::vector<T> c){
    int d = a.size();
    assert(d + 1 == int(c.size()));
    std::vector<T> P = atcoder::convolution(a, c);
    P.resize(d);
    std::vector<T> Q = FPS_Inv_Consecutive(l - d, r, c);
    P = atcoder::convolution(P, Q);
    for (int i = 0; i < r - l; i++){
        P[i] = P[i + d];
    }
    P.resize(r - l);
    return P;
}
} // namespace po167
#line 5 "test/fps/consecutive_linear.test.cpp"
#include<iostream>
int main(){
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr);
    using mint = atcoder::modint998244353;

    long long d, k, M, num;
    std::cin >> d >> k >> M;
    std::vector<mint> A(d), C(d + 1);
    for (int i = 0; i < d; i++) std::cin >> num, A[i] = num;
    for (int i = 1; i <= d; i++) std::cin >> num, C[i] = num;
    C[0] = -1;
    auto ans = po167::Consecutive_Linear(k, k + M, A, C);
    for (int i = 0; i < (int)ans.size(); i++){
        if (i) std::cout << " ";
        std::cout << ans[i].val();
    }
    std::cout << "\n";
}
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