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#define PROBLEM \ "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence" #include <iostream> #include <vector> #include "../../convolution/ntt.hpp" #include "../../math/bostan_mori_algorithm.hpp" #include "../../modint/modint.hpp" using mint = ebi::modint998244353; int main() { int d; long long k; std::cin >> d >> k; std::vector<mint> a(d), c(d); for (int i = 0; i < d; i++) { int val; std::cin >> val; a[i] = val; } for (int i = 0; i < d; i++) { int val; std::cin >> val; c[i] = val; } std::cout << ebi::kitamasa<mint, ebi::convolution>(k, a, c).val() << '\n'; }
#line 1 "test/math/Kth_term_of_Linearly_Recurrent_Sequence.test.cpp" #define PROBLEM \ "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence" #include <iostream> #include <vector> #line 2 "convolution/ntt.hpp" #include <algorithm> #include <array> #include <bit> #include <cassert> #line 8 "convolution/ntt.hpp" #line 2 "math/internal_math.hpp" #line 4 "math/internal_math.hpp" namespace ebi { namespace internal { constexpr int primitive_root_constexpr(int m) { if (m == 2) return 1; if (m == 167772161) return 3; if (m == 469762049) return 3; if (m == 754974721) return 11; if (m == 998244353) return 3; if (m == 880803841) return 26; if (m == 924844033) return 5; return -1; } template <int m> constexpr int primitive_root = primitive_root_constexpr(m); } // namespace internal } // namespace ebi #line 2 "modint/base.hpp" #include <concepts> #line 5 "modint/base.hpp" #include <utility> namespace ebi { template <class T> concept Modint = requires(T a, T b) { a + b; a - b; a * b; a / b; a.inv(); a.val(); a.pow(std::declval<long long>()); T::mod(); }; template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) { long long x; os >> x; a = x; return os; } template <Modint mint> std::ostream &operator<<(std::ostream &os, const mint &a) { return os << a.val(); } } // namespace ebi #line 11 "convolution/ntt.hpp" namespace ebi { namespace internal { template <Modint mint, int g = internal::primitive_root<mint::mod()>> struct ntt_info { static constexpr int rank2 = std::countr_zero((unsigned int)(mint::mod() - 1)); std::array<mint, rank2 + 1> root, inv_root; ntt_info() { root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2); inv_root[rank2] = root[rank2].inv(); for (int i = rank2 - 1; i >= 0; i--) { root[i] = root[i + 1] * root[i + 1]; inv_root[i] = inv_root[i + 1] * inv_root[i + 1]; } } }; template <Modint mint> void butterfly(std::vector<mint>& a) { static const ntt_info<mint> info; int n = int(a.size()); int bit_size = std::countr_zero(a.size()); assert(n == (int)std::bit_ceil(a.size())); // bit reverse for (int i = 0, j = 1; j < n - 1; j++) { for (int k = n >> 1; k > (i ^= k); k >>= 1) ; if (j < i) { std::swap(a[i], a[j]); } } for (int bit = 0; bit < bit_size; bit++) { for (int i = 0; i < n / (1 << (bit + 1)); i++) { mint zeta1 = 1; mint zeta2 = info.root[1]; for (int j = 0; j < (1 << bit); j++) { int idx = i * (1 << (bit + 1)) + j; int jdx = idx + (1 << bit); mint p1 = a[idx]; mint p2 = a[jdx]; a[idx] = p1 + zeta1 * p2; a[jdx] = p1 + zeta2 * p2; zeta1 *= info.root[bit + 1]; zeta2 *= info.root[bit + 1]; } } } } template <Modint mint> void butterfly_inv(std::vector<mint>& a) { static const ntt_info<mint> info; int n = int(a.size()); int bit_size = std::countr_zero(a.size()); assert(n == (int)std::bit_ceil(a.size())); // bit reverse for (int i = 0, j = 1; j < n - 1; j++) { for (int k = n >> 1; k > (i ^= k); k >>= 1) ; if (j < i) { std::swap(a[i], a[j]); } } for (int bit = 0; bit < bit_size; bit++) { for (int i = 0; i < n / (1 << (bit + 1)); i++) { mint zeta1 = 1; mint zeta2 = info.inv_root[1]; for (int j = 0; j < (1 << bit); j++) { int idx = i * (1 << (bit + 1)) + j; int jdx = idx + (1 << bit); mint p1 = a[idx]; mint p2 = a[jdx]; a[idx] = p1 + zeta1 * p2; a[jdx] = p1 + zeta2 * p2; zeta1 *= info.inv_root[bit + 1]; zeta2 *= info.inv_root[bit + 1]; } } } mint inv_n = mint(n).inv(); for (int i = 0; i < n; i++) { a[i] *= inv_n; } } } // namespace internal template <Modint mint> std::vector<mint> convolution_naive(const std::vector<mint>& f, const std::vector<mint>& g) { if (f.empty() || g.empty()) return {}; int n = int(f.size()), m = int(g.size()); std::vector<mint> c(n + m - 1); if (n < m) { for (int j = 0; j < m; j++) { for (int i = 0; i < n; i++) { c[i + j] += f[i] * g[j]; } } } else { for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { c[i + j] += f[i] * g[j]; } } } return c; } template <Modint mint> std::vector<mint> convolution(const std::vector<mint>& f, const std::vector<mint>& g) { if (f.empty() || g.empty()) return {}; if (std::min(f.size(), g.size()) < 60) return convolution_naive(f, g); int n = std::bit_ceil(f.size() + g.size() - 1); std::vector<mint> a(n), b(n); std::copy(f.begin(), f.end(), a.begin()); std::copy(g.begin(), g.end(), b.begin()); internal::butterfly(a); internal::butterfly(b); for (int i = 0; i < n; i++) { a[i] *= b[i]; } internal::butterfly_inv(a); a.resize(f.size() + g.size() - 1); return a; } } // namespace ebi #line 2 "math/bostan_mori_algorithm.hpp" #include <cstdint> #line 5 "math/bostan_mori_algorithm.hpp" namespace ebi { template <class T, std::vector<T> (*convolution)(const std::vector<T> &, const std::vector<T> &)> T bostan_mori_algorithm(std::int64_t n, std::vector<T> p, std::vector<T> q) { while (n > 0) { auto q_neg = q; for (int i = 1; i < (int)q_neg.size(); i += 2) q_neg[i] = -q_neg[i]; p = convolution(p, q_neg); q = convolution(q, q_neg); for (int i = (n & 1LL); i < (int)p.size(); i += 2) p[i >> 1] = p[i]; p.resize(((int)p.size() + 1 - (n & 1LL)) / 2); for (int i = 0; i < (int)q.size(); i += 2) q[i >> 1] = q[i]; q.resize(((int)q.size() + 1) / 2); n >>= 1; } return p[0] / q[0]; } template <class T, std::vector<T> (*convolution)(const std::vector<T> &, const std::vector<T> &)> T kitamasa(std::int64_t n, std::vector<T> a, std::vector<T> c) { if (n < (int)a.size()) return a[n]; const int d = c.size(); for (auto &val : c) val = -val; c.insert(c.begin(), 1); auto p = convolution(a, c); p.resize(d); return bostan_mori_algorithm<T, convolution>(n, p, c); } } // namespace ebi #line 2 "modint/modint.hpp" #line 5 "modint/modint.hpp" #line 7 "modint/modint.hpp" namespace ebi { template <int m> struct static_modint { private: using modint = static_modint; public: static constexpr int mod() { return m; } static constexpr modint raw(int v) { modint x; x._v = v; return x; } constexpr static_modint() : _v(0) {} constexpr static_modint(long long v) { v %= (long long)umod(); if (v < 0) v += (long long)umod(); _v = (unsigned int)v; } constexpr unsigned int val() const { return _v; } constexpr unsigned int value() const { return val(); } constexpr modint &operator++() { _v++; if (_v == umod()) _v = 0; return *this; } constexpr modint &operator--() { if (_v == 0) _v = umod(); _v--; return *this; } constexpr modint operator++(int) { modint res = *this; ++*this; return res; } constexpr modint operator--(int) { modint res = *this; --*this; return res; } constexpr modint &operator+=(const modint &rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } constexpr modint &operator-=(const modint &rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } constexpr modint &operator*=(const modint &rhs) { unsigned long long x = _v; x *= rhs._v; _v = (unsigned int)(x % (unsigned long long)umod()); return *this; } constexpr modint &operator/=(const modint &rhs) { return *this = *this * rhs.inv(); } constexpr modint operator+() const { return *this; } constexpr modint operator-() const { return modint() - *this; } constexpr modint pow(long long n) const { assert(0 <= n); modint x = *this, res = 1; while (n) { if (n & 1) res *= x; x *= x; n >>= 1; } return res; } constexpr modint inv() const { assert(_v); return pow(umod() - 2); } friend modint operator+(const modint &lhs, const modint &rhs) { return modint(lhs) += rhs; } friend modint operator-(const modint &lhs, const modint &rhs) { return modint(lhs) -= rhs; } friend modint operator*(const modint &lhs, const modint &rhs) { return modint(lhs) *= rhs; } friend modint operator/(const modint &lhs, const modint &rhs) { return modint(lhs) /= rhs; } friend bool operator==(const modint &lhs, const modint &rhs) { return lhs.val() == rhs.val(); } friend bool operator!=(const modint &lhs, const modint &rhs) { return !(lhs == rhs); } private: unsigned int _v = 0; static constexpr unsigned int umod() { return m; } }; using modint998244353 = static_modint<998244353>; using modint1000000007 = static_modint<1000000007>; } // namespace ebi #line 10 "test/math/Kth_term_of_Linearly_Recurrent_Sequence.test.cpp" using mint = ebi::modint998244353; int main() { int d; long long k; std::cin >> d >> k; std::vector<mint> a(d), c(d); for (int i = 0; i < d; i++) { int val; std::cin >> val; a[i] = val; } for (int i = 0; i < d; i++) { int val; std::cin >> val; c[i] = val; } std::cout << ebi::kitamasa<mint, ebi::convolution>(k, a, c).val() << '\n'; }