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#include "convolution/convolution.hpp"
$a$ と $b$ を畳み込み、その配列を返す。
愚直に畳み込む。 $O(N^2)$
NTT friendlyな素数における $O(N\log N)$ 時間での畳み込み。 $O(N\log N)$
#pragma once #include <algorithm> #include <bit> #include <vector> #include "../convolution/ntt.hpp" #include "../modint/base.hpp" namespace ebi { 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 = (int)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::fft4(a); internal::fft4(b); for (int i = 0; i < n; i++) { a[i] *= b[i]; } internal::ifft4(a); a.resize(f.size() + g.size() - 1); mint inv_n = mint(n).inv(); for (auto& x : a) x *= inv_n; return a; } } // namespace ebi
#line 2 "convolution/convolution.hpp" #include <algorithm> #include <bit> #include <vector> #line 2 "convolution/ntt.hpp" #line 4 "convolution/ntt.hpp" #include <array> #line 6 "convolution/ntt.hpp" #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> #include <iostream> #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 2 "template/int_alias.hpp" #include <cstdint> namespace ebi { using ld = long double; using std::size_t; using i8 = std::int8_t; using u8 = std::uint8_t; using i16 = std::int16_t; using u16 = std::uint16_t; using i32 = std::int32_t; using u32 = std::uint32_t; using i64 = std::int64_t; using u64 = std::uint64_t; using i128 = __int128_t; using u128 = __uint128_t; } // namespace ebi #line 12 "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 fft2(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 == 1 << bit_size); for (int bit = bit_size - 1; bit >= 0; bit--) { int m = 1 << bit; for (int i = 0; i < n; i += 2 * m) { mint w = 1; for (int j = 0; j < m; j++) { mint p1 = a[i + j]; mint p2 = a[i + j + m]; a[i + j] = p1 + p2; a[i + j + m] = (p1 - p2) * w; w *= info.root[bit + 1]; } } } } template <Modint mint> void ifft2(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 == 1 << bit_size); for (int bit = 0; bit < bit_size; bit++) { for (int i = 0; i < n / (1 << (bit + 1)); i++) { mint w = 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 = w * a[jdx]; a[idx] = p1 + p2; a[jdx] = p1 - p2; w *= info.inv_root[bit + 1]; } } } } template <Modint mint> void fft4(std::vector<mint>& a) { static const ntt_info<mint> info; const u32 mod = mint::mod(); const u64 iw = info.root[2].val(); int n = int(a.size()); int bit_size = std::countr_zero(a.size()); assert(n == 1 << bit_size); int len = bit_size; while (len > 0) { if (len == 1) { for (int i = 0; i < n; i += 2) { mint p0 = a[i]; mint p1 = a[i + 1]; a[i] = p0 + p1; a[i + 1] = p0 - p1; } len--; } else { int m = 1 << (len - 2); u64 w1 = 1, w2 = 1, w3 = 1, iw1 = iw, iw3 = iw; for (int i = 0; i < m; i++) { for (int j = 0; j < n; j += 4 * m) { int i0 = i + j, i1 = i0 + m, i2 = i1 + m, i3 = i2 + m; u32 a0 = a[i0].val(); u32 a1 = a[i1].val(); u32 a2 = a[i2].val(); u32 a3 = a[i3].val(); u32 a0_plus_a2 = a0 + a2; u32 a1_plus_a3 = a1 + a3; u32 a0_minus_a2 = a0 + mod - a2; u32 a1_minus_a3 = a1 + mod - a3; a[i0] = a0_plus_a2 + a1_plus_a3; a[i1] = a0_minus_a2 * w1 + a1_minus_a3 * iw1; a[i2] = (a0_plus_a2 + 2 * mod - a1_plus_a3) * w2; a[i3] = a0_minus_a2 * w3 + (2 * mod - a1_minus_a3) * iw3; } w1 = w1 * info.root[len].val() % mod; w2 = w1 * w1 % mod; w3 = w2 * w1 % mod; iw1 = iw * w1 % mod; iw3 = iw * w3 % mod; } len -= 2; } } } template <Modint mint> void ifft4(std::vector<mint>& a) { static const ntt_info<mint> info; const u32 mod = mint::mod(); const u64 mod2 = u64(mod) * mod; const u64 iw = info.inv_root[2].val(); int n = int(a.size()); int bit_size = std::countr_zero(a.size()); assert(n == 1 << bit_size); int len = (bit_size & 1 ? 1 : 2); while (len <= bit_size) { if (len == 1) { for (int i = 0; i < n; i += 2) { mint a0 = a[i]; mint a1 = a[i + 1]; a[i] = a0 + a1; a[i + 1] = a0 - a1; } } else { int m = 1 << (len - 2); u64 w1 = 1, w2 = 1, w3 = 1, iw1 = iw, iw3 = iw; for (int i = 0; i < m; i++) { for (int j = 0; j < n; j += 4 * m) { int i0 = i + j, i1 = i0 + m, i2 = i1 + m, i3 = i2 + m; u64 a0 = a[i0].val(); u64 a1 = w1 * a[i1].val(); u64 a2 = w2 * a[i2].val(); u64 a3 = w3 * a[i3].val(); u64 b1 = iw1 * a[i1].val(); u64 b3 = iw3 * a[i3].val(); u64 a0_plus_a2 = a0 + a2; u64 a1_plus_a3 = a1 + a3; u64 a0_minus_a2 = a0 + mod2 - a2; u64 b1_minus_b3 = b1 + mod2 - b3; a[i0] = a0_plus_a2 + a1_plus_a3; a[i1] = a0_minus_a2 + b1_minus_b3; a[i2] = a0_plus_a2 + mod2 * 2 - a1_plus_a3; a[i3] = a0_minus_a2 + mod2 * 2 - b1_minus_b3; } w1 = w1 * info.inv_root[len].val() % mod; w2 = w1 * w1 % mod; w3 = w2 * w1 % mod; iw1 = iw * w1 % mod; iw3 = iw * w3 % mod; } } len += 2; } } } // namespace internal } // namespace ebi #line 9 "convolution/convolution.hpp" namespace ebi { 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 = (int)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::fft4(a); internal::fft4(b); for (int i = 0; i < n; i++) { a[i] *= b[i]; } internal::ifft4(a); a.resize(f.size() + g.size() - 1); mint inv_n = mint(n).inv(); for (auto& x : a) x *= inv_n; return a; } } // namespace ebi