FFT模板

FFT 实际是 DFT 的一种快速实现方法

可以将多项式的乘法从 O(n^2) 优化到 O(nlogn)

暂时没有看到很好的科普文章、原理自行百度吧

#define L(x) (1 << (x))
const double PI = acos(-1.0);
const int maxn = (1<<17) + (int)1e3;
double ax[maxn], ay[maxn], bx[maxn], by[maxn];
int revv(int x, int bits)
{
    int ret = 0;
    for (int i = 0; i < bits; i++){
        ret <<= 1;
        ret |= x & 1;
        x >>= 1;
    }
    return ret;
}

void fft(double * a, double * b, int n, bool rev)
{
    int bits = 0;
    while (1 << bits < n) ++bits;
    for (int i = 0; i < n; i++){
        int j = revv(i, bits);
        if (i < j) swap(a[i], a[j]), swap(b[i], b[j]);
    }

    for (int len = 2; len <= n; len <<= 1){
        int half = len >> 1;
        double wmx = cos(2 * PI / len), wmy = sin(2 * PI / len);
        if (rev) wmy = -wmy;
        for (int i = 0; i < n; i += len){
            double wx = 1, wy = 0;
            for (int j = 0; j < half; j++){
                double cx = a[i + j], cy = b[i + j];
                double dx = a[i + j + half], dy = b[i + j + half];
                double ex = dx * wx - dy * wy, ey = dx * wy + dy * wx;
                a[i + j] = cx + ex, b[i + j] = cy + ey;
                a[i + j + half] = cx - ex, b[i + j + half] = cy - ey;
                double wnx = wx * wmx - wy * wmy, wny = wx * wmy + wy * wmx;
                wx = wnx, wy = wny;
            }
        }
    }
    if (rev){
        for (int i = 0; i < n; i++)
            a[i] /= n, b[i] /= n;
    }
}

int Convolution(int a[],int na,int b[],int nb,int ans[]) //两个数组求卷积,有时ans数组要开成long long
{
    int len = max(na, nb), ln;
    for(ln=0; L(ln)<len; ++ln);
    len=L(++ln);
    for (int i = 0; i < len ; ++i){
        if (i >= na) ax[i] = 0, ay[i] =0;
        else ax[i] = a[i], ay[i] = 0;
    }
    fft(ax, ay, len, 0);
    for (int i = 0; i < len; ++i){
        if (i >= nb) bx[i] = 0, by[i] = 0;
        else bx[i] = b[i], by[i] = 0;
    }
    fft(bx, by, len, 0);
    for (int i = 0; i < len; ++i){
        double cx = ax[i] * bx[i] - ay[i] * by[i];
        double cy = ax[i] * by[i] + ay[i] * bx[i];
        ax[i] = cx, ay[i] = cy;
    }
    fft(ax, ay, len, 1);
    for (int i = 0; i < len; ++i)
        ans[i] = (int)(ax[i] + 0.5);
    return len;
}


int Convolution_self(long long a[], int na, int ans[]) //自己跟自己求卷积,有时候ans数组要开成long long
{
    int len = na, ln;
    for(ln = 0; L(ln) < na; ++ln);
    len=L(++ln);
    for(int i = 0; i < len; ++i){
        if (i >= na) ax[i] = 0, ay[i] = 0;
        else ax[i] = a[i], ay[i] = 0;
    }
    fft(ax, ay, len, 0);
    for(int i=0; i<len; ++i){
        double cx = ax[i] * ax[i] - ay[i] * ay[i];
        double cy = 2 * ax[i] * ay[i];
        ax[i] = cx, ay[i] = cy;
    }
    fft(ax, ay, len, 1);

    for(int i=0; i<len; ++i)
        ans[i] = ax[i] + 0.5;
    return len;
}

模板题

#include<bits/stdc++.h>
#define LL long long
#define ULL unsigned long long

#define scl(i) scanf("%lld", &i)
#define scll(i, j) scanf("%lld %lld", &i, &j)
#define sclll(i, j, k) scanf("%lld %lld %lld", &i, &j, &k)
#define scllll(i, j, k, l) scanf("%lld %lld %lld %lld", &i, &j, &k, &l)

#define scs(i) scanf("%s", i)
#define sci(i) scanf("%d", &i)
#define scd(i) scanf("%lf", &i)
#define scIl(i) scanf("%I64d", &i)
#define scii(i, j) scanf("%d %d", &i, &j)
#define scdd(i, j) scanf("%lf %lf", &i, &j)
#define scIll(i, j) scanf("%I64d %I64d", &i, &j)
#define sciii(i, j, k) scanf("%d %d %d", &i, &j, &k)
#define scddd(i, j, k) scanf("%lf %lf %lf", &i, &j, &k)
#define scIlll(i, j, k) scanf("%I64d %I64d %I64d", &i, &j, &k)
#define sciiii(i, j, k, l) scanf("%d %d %d %d", &i, &j, &k, &l)
#define scdddd(i, j, k, l) scanf("%lf %lf %lf %lf", &i, &j, &k, &l)
#define scIllll(i, j, k, l) scanf("%I64d %I64d %I64d %I64d", &i, &j, &k, &l)

#define lson l, m, rt<<1
#define rson m+1, r, rt<<1|1
#define lowbit(i) (i & (-i))
#define mem(i, j) memset(i, j, sizeof(i))

#define fir first
#define sec second
#define VI vector<int>
#define ins(i) insert(i)
#define pb(i) push_back(i)
#define pii pair<int, int>
#define VL vector<long long>
#define mk(i, j) make_pair(i, j)
#define all(i) i.begin(), i.end()
#define pll pair<long long, long long>

#define _TIME 0
#define _INPUT 0
#define _OUTPUT 0
clock_t START, END;
void __stTIME();
void __enTIME();
void __IOPUT();
using namespace std;

#define L(x) (1 << (x))
const double PI = acos(-1.0);
const int maxn = (1<<20) + (int)1e3;
double ax[maxn], ay[maxn], bx[maxn], by[maxn];
int revv(int x, int bits)
{
    int ret = 0;
    for (int i = 0; i < bits; i++){
        ret <<= 1;
        ret |= x & 1;
        x >>= 1;
    }
    return ret;
}

void fft(double * a, double * b, int n, bool rev)
{
    int bits = 0;
    while (1 << bits < n) ++bits;
    for (int i = 0; i < n; i++){
        int j = revv(i, bits);
        if (i < j) swap(a[i], a[j]), swap(b[i], b[j]);
    }

    for (int len = 2; len <= n; len <<= 1){
        int half = len >> 1;
        double wmx = cos(2 * PI / len), wmy = sin(2 * PI / len);
        if (rev) wmy = -wmy;
        for (int i = 0; i < n; i += len){
            double wx = 1, wy = 0;
            for (int j = 0; j < half; j++){
                double cx = a[i + j], cy = b[i + j];
                double dx = a[i + j + half], dy = b[i + j + half];
                double ex = dx * wx - dy * wy, ey = dx * wy + dy * wx;
                a[i + j] = cx + ex, b[i + j] = cy + ey;
                a[i + j + half] = cx - ex, b[i + j + half] = cy - ey;
                double wnx = wx * wmx - wy * wmy, wny = wx * wmy + wy * wmx;
                wx = wnx, wy = wny;
            }
        }
    }
    if (rev){
        for (int i = 0; i < n; i++)
            a[i] /= n, b[i] /= n;
    }
}

int Convolution(int a[],int na,int b[],int nb,int ans[]) //两个数组求卷积,有时ans数组要开成long long
{
    int len = max(na, nb), ln;
    for(ln=0; L(ln)<len; ++ln);
    len=L(++ln);
    for (int i = 0; i < len ; ++i){
        if (i >= na) ax[i] = 0, ay[i] =0;
        else ax[i] = a[i], ay[i] = 0;
    }
    fft(ax, ay, len, 0);
    for (int i = 0; i < len; ++i){
        if (i >= nb) bx[i] = 0, by[i] = 0;
        else bx[i] = b[i], by[i] = 0;
    }
    fft(bx, by, len, 0);
    for (int i = 0; i < len; ++i){
        double cx = ax[i] * bx[i] - ay[i] * by[i];
        double cy = ax[i] * by[i] + ay[i] * bx[i];
        ax[i] = cx, ay[i] = cy;
    }
    fft(ax, ay, len, 1);
    for (int i = 0; i < len; ++i)
        ans[i] = (int)(ax[i] + 0.5);
    return len;
}


int Convolution_self(long long a[], int na, int ans[]) //自己跟自己求卷积,有时候ans数组要开成long long
{
    int len = na, ln;
    for(ln = 0; L(ln) < na; ++ln);
    len=L(++ln);
    for(int i = 0; i < len; ++i){
        if (i >= na) ax[i] = 0, ay[i] = 0;
        else ax[i] = a[i], ay[i] = 0;
    }
    fft(ax, ay, len, 0);
    for(int i=0; i<len; ++i){
        double cx = ax[i] * ax[i] - ay[i] * ay[i];
        double cy = 2 * ax[i] * ay[i];
        ax[i] = cx, ay[i] = cy;
    }
    fft(ax, ay, len, 1);

    for(int i=0; i<len; ++i)
        ans[i] = ax[i] + 0.5;
    return len;
}

int a[maxn], b[maxn], c[maxn];
int main(void){__stTIME();__IOPUT();

    int n, m;
    scii(n, m);

    for(int i=0; i<=n; i++) sci(a[i]);
    for(int i=0; i<=m; i++) sci(b[i]);

    int len = Convolution(a, n+1, b, m+1, c);

    for(int i=0; i<n+m+1; i++) printf("%d ", c[i]); puts("");


__enTIME();return 0;}


void __stTIME()
{
    #if _TIME
        START = clock();
    #endif
}

void __enTIME()
{
    #if _TIME
        END = clock();
        cerr<<"execute time = "<<(double)(END-START)/CLOCKS_PER_SEC<<endl;
    #endif
}

void __IOPUT()
{
    #if _INPUT
        freopen("in.txt", "r", stdin);
    #endif
    #if _OUTPUT
        freopen("out.txt", "w", stdout);
    #endif
}