浅谈FWT

这种东西直接背板子就好了

感觉如果懂FFT的话FWT还是很好理解的

也是转换为点值表示法（FWT）然后对应位乘起来就好了
然后再IFWT就可以了

YYB dalao写的FWT太好了！！

像我就直接背板子

$A_0是二进制开头为0的，A_1是二进制开头为1的$

or

如果卷积长这样 $C_k={\sum }_{i|j=k}{A}_i\ast B_j$

$FWT(A)=(FWT(A_0),FWT(A_0+A_1))$

$IFWT(A)=(FWT(A_0),FWT(A_0-A_1))$

and

如果卷积长这样 $C_k={\sum }_{i\&j=k}{A}_i\ast B_j$

$FWT(A)=(FWT(A_0+A_1),FWT(A_1))$

$IFWT(A)=(FWT(A_0-A_1),FWT(A_1))$

xor

如果卷积长这样 $C_k={\sum }_{i\oplus j=k}{A}_i\ast B_j$

$FWT(A)=(FWT(A_0+A_1),FWT(A_0-A_1))$

$IFWT(A)=(FWT(A_0+A_1)/2,FWT(A_0-A_1)/2)$

然后就和FFT一样用就行了

模板题

code:

#include<bits/stdc++.h>
#define mod 998244353
#define ll long long
using namespace std;
const int N = (1 << 17) + 5;
void fwt_or(ll *a, int len, int opt){
	for(int i = 2; i <= len; i <<= 1)
		for(int p = i >> 1, j = 0; j + i <= len; j += i)
			for(int k = j; k < j + p; k ++)
				a[p + k] =(a[p + k] + opt * a[k] + mod) % mod;
}
void fwt_and(ll *a, int len, int opt){
	for(int i = 2; i <= len; i <<= 1)
		for(int p = i >> 1, j = 0; j + i <= len; j += i)
			for(int k = j; k < j + p; k ++)
				a[k] = (a[k] + opt * a[k + p] + mod) % mod;
}
 
void fwt_xor(ll *a, int len, int opt){
	for(int i = 2; i <= len; i <<= 1)
		for(int p = i >> 1, j = 0; j + i <= len; j += i)
			for(int k = j; k < j + p; k ++){
				int X = a[k], Y = a[k + p];
				a[k] = (X + Y) % mod, a[k + p] = (X - Y + mod) % mod;
				if(opt == -1) a[k] = a[k] * 499122177 % mod, a[k + p] = a[k + p] * 499122177 % mod;
			}
				
}

int n;
ll a[N], b[N], f[N], g[N];
int main(){
	scanf("%d", &n);
	int len = (1 << n);
	for(int i = 0; i < len; i ++) scanf("%lld", &a[i]);
	for(int i = 0; i < len; i ++) scanf("%lld", &b[i]);
	
	for(int i = 0; i < len; i ++) f[i] = a[i], g[i] = b[i];
	fwt_or(f, len, 1), fwt_or(g, len, 1);
	for(int i = 0; i < len; i ++) f[i] = f[i] * g[i] % mod;
	fwt_or(f, len, -1);
	for(int i = 0; i < len; i ++) printf("%lld ", f[i]); printf("\n");
	
	for(int i = 0; i < len; i ++) f[i] = a[i], g[i] = b[i];
	fwt_and(f, len, 1), fwt_and(g, len, 1);
	for(int i = 0; i < len; i ++) f[i] = f[i] * g[i] % mod;
	fwt_and(f, len, -1);
	for(int i = 0; i < len; i ++) printf("%lld ", f[i]); printf("\n");
	
	for(int i = 0; i < len; i ++) f[i] = a[i], g[i] = b[i];
	fwt_xor(f, len, 1), fwt_xor(g, len, 1);
	for(int i = 0; i < len; i ++) f[i] = f[i] * g[i] % mod;
	fwt_xor(f, len, -1);
	for(int i = 0; i < len; i ++) printf("%lld ", f[i]); printf("\n");
	
	
	return 0;
}

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