拆系数FFT(任意模数FFT)

拆系数FFT

对于任意模数 \(mod\)

设\(m=\sqrt {mod}\)

把多项式\(A(x)\)和\(B(x)\)的系数都拆成\(a\times m+b\)的形式，时\(a, b\)都小于\(m\)

提出，那么一个多项式就可以拆成两个多项式的加法

一个是\(a*m\)的，一个是\(b\)的

直接乘法分配律，\(aa\)一遍，\(ab\)一遍，\(ba\)，\(bb\)一遍，四遍\(FFT\)

乘出来不会超过取模范围

然后合并直接

\[(a\times m+b)(c\times m+d)=a\times c\times m^2+(a\times c+b\times d)m+b\times d
\]

这样子要进行 \(7\) 次 \(DFT\)

如果研究一下 \(myy\) \(2016\) 年的集训队论文就会发现有 \(2\) 次 或者 \(1.5\) 次 \(DFT\) 的 \(FFT\) 算法

2次的够了吧

\(myy\) 巧妙的运用了复数的虚部，优化了算法

具体来说

设

\[C(x)=A(x)+iB(x)
\]

\[D(x)=A(x)-iB(x)
\]

假设

\(c(w_n^k)\) 表示将 \(C(x)\) \(DFT\) 后的点值

\(d(w_n^k)\) 表示将 \(D(x)\) \(DFT\) 后的点值

\(w\) 为 \(n\) 次单位复数根

设 \(conj(x)\) 表示 \(x\) 的共轭复数

那么

\[c(w_{2n}^{k})=A(w_{2n}^{k})+iB(w_{2n}^{k})=\sum_{j=0}^{2n-1}A_jw_{2n}^{jk}+iB_jw_{2n}^{jk}
\]

\[=\sum_{j=0}^{2n-1}(A_j+iB_j)(cos\frac{\pi jk}{n}+isin \frac{\pi jk}{n})
\]

\[d(w_{2n}^{k})=A(w_{2n}^{k})-iB(w_{2n}^{k})=\sum_{j=0}^{2n-1}(A_j-iB_j)(cos\frac{\pi jk}{n}+isin \frac{\pi jk}{n})
\]

设 \(x=\frac{\pi jk}{n}\)

那么

\[d(w_{2n}^{k})=\sum_{j=0}^{2n-1}(A_jcosx+B_jsinx)+i(A_jsinx-B_jcosx)
\]

\[=conj(\sum_{j=0}^{2n-1}(A_jcosx+B_jsinx)-i(A_jsinx-B_jcosx))
\]

\[=conj(\sum_{j=0}^{2n-1}(A_jcos(-x)-B_jsin(-x))+i(A_jsin(-x)+B_jcos(-x)))
\]

\[=conj(\sum_{j=0}^{2n-1}(A_j+iB_j)(cos(-x)+isin(-x)))
\]

\[=conj(\sum_{j=0}^{2n-1}(A_j+iB_j)(cos(2\pi-x)+isin(2\pi-x)))
\]

\[=conj(c(w_{2n}^{2n-k}))
\]

也就是说

只需要一次 \(DFT\) 求出 \(c\) 就可以求出 \(d\)

那么

\[DFT(A(k))=\frac{c(w^{k})+d(w^{k})}{2}
\]

\[DFT(B(k))=\frac{c(w^{k})-d(w^{k})}{2i}
\]

再用一次 \(DFT^{-1}\) 还原出多项式，多项式乘法只要两次 \(DFT\)

考虑拆系数 \(FFT\)

\(DFT\) 两两合并，从 \(4\) 次变成 \(2\) 次

\(DFT^{-1}\) 合并其中一个，从 \(3\) 次变成 \(2\) 次

一共 \(4\) 次

# include <bits/stdc++.h>
using namespace std;
typedef long long ll;

namespace IO {
    const int maxn((1 << 21) + 1);

    char ibuf[maxn], *iS, *iT, obuf[maxn], *oS = obuf, *oT = obuf + maxn - 1, c, st[65];
    int f, tp;

    char Getc() {
        return (iS == iT ? (iT = (iS = ibuf) + fread(ibuf, 1, maxn, stdin), (iS == iT ? EOF : *iS++)) : *iS++);
    }

    void Flush() {
        fwrite(obuf, 1, oS - obuf, stdout);
        oS = obuf;
    }

    void Putc(char x) {
        *oS++ = x;
        if (oS == oT) Flush();
    }

    template <class Int> void In(Int &x) {
        for (f = 1, c = Getc(); c < '0' || c > '9'; c = Getc()) f = c == '-' ? -1 : 1;
        for (x = 0; c <= '9' && c >= '0'; c = Getc()) x = (x << 3) + (x << 1) + (c ^ 48);
        x *= f;
    }

    template <class Int> void Out(Int x) {
        if (!x) Putc('0');
        if (x < 0) Putc('-'), x = -x;
        while (x) st[++tp] = x % 10 + '0', x /= 10;
        while (tp) Putc(st[tp--]);
    }
}

using IO :: In;
using IO :: Out;
using IO :: Putc;
using IO :: Flush;

const int maxn(1 << 18);
const double pi(acos(-1));

struct Complex {
    double a, b;

    inline Complex() {
        a = b = 0;
    }

    inline Complex(double _a, double _b) {
        a = _a, b = _b;
    }

    inline Complex operator +(Complex x) const {
        return Complex(a + x.a, b + x.b);
    }

    inline Complex operator -(Complex x) const {
        return Complex(a - x.a, b - x.b);
    }

    inline Complex operator *(Complex x) const {
        return Complex(a * x.a - b * x.b, a * x.b + b * x.a);
    }

    inline Complex Conj() {
        return Complex(a, -b);
    }
};

Complex a[maxn], b[maxn], w[maxn], a1[maxn], a2[maxn];
int r[maxn], l, deg, g[maxn], h[maxn], mod;

inline void FFT(Complex *p, int opt) {
    register int i, j, k, t;
    register Complex wn, x, y;
    for (i = 0; i < deg; ++i) if (r[i] < i) swap(p[r[i]], p[i]);
    for (i = 1; i < deg; i <<= 1)
        for(t = i << 1, j = 0; j < deg; j += t)
            for (k = 0; k < i; ++k) {
                wn = w[deg / i * k];
                if (opt == -1) wn.b *= -1;
                x = p[j + k], y = wn * p[i + j + k];
                p[j + k] = x + y, p[i + j + k] = x - y;
            }
}

inline void Mul(int n, int *p, int *q, int *f) {
    register int i, k, v1, v2, v3;
    register Complex ca, cb, da1, da2, db1, db2;
    for (deg = 1, l = 0; deg < n; deg <<= 1) ++l;
    for (i = 0; i < deg; ++i) r[i] = (r[i >> 1] >> 1) | ((i & 1) << (l - 1));
    for (i = 0; i < deg; ++i) w[i] = Complex(cos(pi * i / deg), sin(pi * i / deg));
    for (i = 0; i < n; ++i) a[i] = Complex(p[i] & 32767, p[i] >> 15), b[i] = Complex(q[i] & 32767, q[i] >> 15);
    for (FFT(a, 1), FFT(b, 1), i = 0; i < deg; ++i) {
        k = (deg - i) & (deg - 1), ca = a[k].Conj(), cb = b[k].Conj();
        da1 = (ca + a[i]) * Complex(0.5, 0), da2 = (a[i] - ca) * Complex(0, -0.5);
        db1 = (cb + b[i]) * Complex(0.5, 0), db2 = (b[i] - cb) * Complex(0, -0.5);
        a1[i] = da1 * db1 + (da1 * db2 + da2 * db1) * Complex(0, 1), a2[i] = da2 * db2;
    }
    for (FFT(a1, -1), FFT(a2, -1), i = 0; i < deg; ++i) {
        v1 = (ll)(a1[i].a / deg + 0.5) % mod, v2 = (ll)(a1[i].b / deg + 0.5) % mod;
        v3 = (ll)(a2[i].a / deg + 0.5) % mod, f[i] = (((ll)v3 << 30) + ((ll)v2 << 15) + v1) % mod;
        if (f[i] < 0) f[i] += mod;
    }
}

int main() {
    register int len, i, n, m;
    In(n), In(m), In(mod), ++n, ++m;
    for (i = 0; i < n; ++i) In(h[i]), h[i] %= mod;
    for (i = 0; i < m; ++i) In(g[i]), g[i] %= mod;
    for (len = 1, n += m - 1; len < n; len <<= 1);
    for (Mul(len, h, g, g), i = 0; i < n; ++i) Out(g[i]), Putc(' ');
    return Flush(), 0;
}