畳み込み FFT
(library/polynomial/fft/convolution_fft.hpp)

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• Last update: 2026-01-21 11:49:22+09:00
• Include: #include "library/polynomial/fft/convolution_fft.hpp"

畳み込み FFT

できること

• 畳み込みは、 $(1 + 2x + 3x^2) \times (4 + 5x + 6x^2)$の係数を求める
• 畳み込みをFFTで高速に行う
• 演算結果はllroundで四捨五入している
• 誤差なしのためには数論変換NTTが必要

計算量

$O(NlogN)$

使い方

vector<long long> res = convolution_fft(a, b);

Depends on

• 高速フーリエ変換 (library/polynomial/fft/fast_fourier_transform.hpp)

Verified with

• 畳み込みFFTのテスト (tests/polynomial.fft.convolution_fft.test.cpp)

Code

#pragma once
#include "library/polynomial/fft/fast_fourier_transform.hpp"
template <typename R>
vector<long long> convolution_fft(const vector<R> &a, const vector<R> &b) {
    using C = FFT::C;
    int need = (int)a.size() + (int)b.size() - 1, nbase = 1;
    while ((1 << nbase) < need) ++nbase;
    FFT::ensure_base(nbase);
    int sz = 1 << nbase;
    vector<C> fa(sz);
    for (int i = 0; i < sz; i++) {
        double x = i < (int)a.size() ? a[i] : 0;
        double y = i < (int)b.size() ? b[i] : 0;
        fa[i] = C(x, y);
    }
    FFT::fft(fa, sz);
    C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
    for (int i = 0; i <= (sz >> 1); i++) {
        int j = (sz - i) & (sz - 1);
        C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
        fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
        fa[i] = z;
    }
    for (int i = 0; i < (sz >> 1); i++) {
        C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
        C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * FFT::rts[(sz >> 1) + i];
        fa[i] = A0 + A1 * s;
    }
    FFT::fft(fa, sz >> 1);
    vector<long long> ret(need);
    for (int i = 0; i < need; i++) {
        ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
    }
    return ret;
}

#line 2 "library/polynomial/fft/fast_fourier_transform.hpp"
namespace FFT {
using real = double;
struct C {
    real x, y;
    C() : x(0), y(0) {};
    C(real x, real y) : x(x), y(y) {};
    inline C operator+(const C &c) const { return C(x + c.x, y + c.y); }
    inline C operator-(const C &c) const { return C(x - c.x, y - c.y); }
    inline C operator*(const C &c) const {
        return C(x * c.x - y * c.y, x * c.y + y * c.x);
    }
    inline C conj() const { return C(x, -y); }
};
const real PI = acosl(-1);
int base = 1;
vector<C> rts = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};
void ensure_base(int nbase) {
    if (nbase <= base) return;
    rev.resize(1 << nbase);
    rts.resize(1 << nbase);
    for (int i = 0; i < (1 << nbase); i++) {
        rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
    }
    while (base < nbase) {
        real angle = PI * 2.0 / (1 << (base + 1));
        for (int i = 1 << (base - 1); i < (1 << base); i++) {
            rts[i << 1] = rts[i];
            real angle_i = angle * (2 * i + 1 - (1 << base));
            rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i));
        }
        ++base;
    }
}
void fft(vector<C> &a, int n) {
    assert((n & (n - 1)) == 0);
    int zeros = __builtin_ctz(n);
    ensure_base(zeros);
    int shift = base - zeros;
    for (int i = 0; i < n; i++) {
        if (i < (rev[i] >> shift)) swap(a[i], a[rev[i] >> shift]);
    }
    for (int k = 1; k < n; k <<= 1) {
        for (int i = 0; i < n; i += 2 * k) {
            for (int j = 0; j < k; j++) {
                C z = a[i + j + k] * rts[j + k];
                a[i + j + k] = a[i + j] - z;
                a[i + j] = a[i + j] + z;
            }
        }
    }
}
vector<int64_t> multiply(const vector<int> &a, const vector<int> &b) {
    int need = (int)a.size() + (int)b.size() - 1;
    int nbase = 1;
    while ((1 << nbase) < need) ++nbase;
    ensure_base(nbase);
    int sz = 1 << nbase;
    vector<C> fa(sz);
    for (int i = 0; i < sz; ++i) {
        int x = (i < (int)a.size() ? a[i] : 0);
        int y = (i < (int)b.size() ? b[i] : 0);
        fa[i] = C(x, y);
    }
    fft(fa, sz);
    C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
    for (int i = 0; i <= (sz >> 1); i++) {
        int j = (sz - i) & (sz - 1);
        C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
        fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
        fa[i] = z;
    }
    for (int i = 0; i < (sz >> 1); i++) {
        C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
        C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * rts[(sz >> 1) + i];
        fa[i] = A0 + A1 * s;
    }
    fft(fa, sz >> 1);
    vector<int64_t> ret(need);
    for (int i = 0; i < need; i++) {
        ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
    }
    return ret;
}
} // namespace FFT
#line 3 "library/polynomial/fft/convolution_fft.hpp"
template <typename R>
vector<long long> convolution_fft(const vector<R> &a, const vector<R> &b) {
    using C = FFT::C;
    int need = (int)a.size() + (int)b.size() - 1, nbase = 1;
    while ((1 << nbase) < need) ++nbase;
    FFT::ensure_base(nbase);
    int sz = 1 << nbase;
    vector<C> fa(sz);
    for (int i = 0; i < sz; i++) {
        double x = i < (int)a.size() ? a[i] : 0;
        double y = i < (int)b.size() ? b[i] : 0;
        fa[i] = C(x, y);
    }
    FFT::fft(fa, sz);
    C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
    for (int i = 0; i <= (sz >> 1); i++) {
        int j = (sz - i) & (sz - 1);
        C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
        fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
        fa[i] = z;
    }
    for (int i = 0; i < (sz >> 1); i++) {
        C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
        C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * FFT::rts[(sz >> 1) + i];
        fa[i] = A0 + A1 * s;
    }
    FFT::fft(fa, sz >> 1);
    vector<long long> ret(need);
    for (int i = 0; i < need; i++) {
        ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
    }
    return ret;
}

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