【BZOJ】2194: 快速傅立叶之二

http://www.lydsy.com/JudgeOnline/problem.php?id=2194

题意：求$c[k]=\sum_{k<=i<n} a[i]b[i-k], n<=10^5$

#include <bits/stdc++.h>
using namespace std;
struct cp {
	double x, y;
	cp(double _x=0, double _y=0):x(_x),y(_y) {}
	cp operator+(const cp &a) { return cp(x+a.x, y+a.y); }
	cp operator-(const cp &a) { return cp(x-a.x, y-a.y); }
	cp operator*(const cp &a) { return cp(x*a.x-y*a.y, x*a.y+y*a.x); }
};
const int N=400005;
const double pi=acos(-1);
int rev[N];
void dft(cp *a, int n, int flag) {
	static cp A[N], u, v;
	static int i, j, m, mid;
	for(i=0; i<n; ++i) A[rev[i]]=a[i];
	for(i=0; i<n; ++i) a[i]=A[i];
	for(m=2; m<=n; m<<=1) {
		cp wn(cos(pi*2/m), sin(pi*2/m)*flag);
		for(i=0; i<n; i+=m) {
			mid=m>>1; cp w(1);
			for(j=0; j<mid; ++j) {
				u=a[i+j], v=a[i+j+mid]*w;
				a[i+j]=u+v;
				a[i+j+mid]=u-v;
				w=w*wn;
			}
		}
	}
	if(flag==-1) for(i=0; i<n; ++i) a[i].x/=n;
}
inline void init(int &len) {
	static int k, t, j, r; k=1; t=0;
	while(k<len) k<<=1, ++t;
	len=k;
	for(int i=0; i<len; ++i) {
		k=i; j=t; r=0;
		while(j--) r<<=1, r|=k&1, k>>=1;
		rev[i]=r;
	}
}
void fft(int *a, int *b, int *c, int la, int lb) {
	static cp x[N], y[N];
	int len=la+lb-1;
	init(len);
	for(int i=0; i<len; ++i) x[i].x=a[i], x[i].y=0;
	for(int i=0; i<len; ++i) y[i].x=b[i], y[i].y=0;
	dft(x, len, 1); dft(y, len, 1);
	for(int i=0; i<len; ++i) x[i]=x[i]*y[i];
	dft(x, len, -1);
	for(int i=0; i<len; ++i) c[i]=x[i].x+0.5;
}
int x[N], y[N], a[N], n;
int main() {
	scanf("%d", &n);
	for(int i=0; i<n; ++i) scanf("%d%d", &x[i], &y[i]);
	for(int i=0; i<n; ++i) a[i]=x[n-1-i];
	fft(y, a, x, n, n);
	for(int i=0; i<n; ++i) a[i]=x[n-1-i];
	for(int i=0; i<n; ++i) printf("%d\n", a[i]);
	return 0;
}

　　

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复习了下fft...能码出来真是极好的= =可是为什么我的fft辣么慢！

首先是很复杂的代换= =

$$
\begin{align}
c[k]
= & \sum_{k<=i<n} a[i]b[i-k] \\
= & \sum_{0<=i-k<=n-k-1} a[i]b[i-k] \\
\end{align}
$$

令$j=i-k, 则i=j+k$

$$ c[k] =  \sum_{0<=j<=n-k-1} a[j+k]b[j] $$

设 $ c'[n-k-1]=c[k] , T=n-k-1 \Rightarrow k=n-1-T $，有

$$ c'[T] = \sum_{0<=j<=T} a[n-1-(T-j)]b[j] $$

最后再设 $ a'[T-j]=a[n-1-(T-j)] \Rightarrow a'[x]=a[n-1-x] $

然后就是裸的卷积辣= =