Luo Gu P4717 [template] Fast Walsh Transform (FWT)

Well given directly to the formula:

Or convolution:

$ FWT [A] = merge (FWT [A0], FWT [A0] + FWT [A1]) $

$UFWT[A']=merge(UFWT[A0'],UFWT[A1']-UFWT[A0'])$

Convolution:

$ FWT [A] = merge (FWT [A0] + FWT [A1], FWT [A1]) $

$UFWT[A']=merge(UFWT[A0']-UFWT[A1'],UFWT[A1'])$

XOR convolution:

$ FWT [A] = merge (FWT [A0] + FWT [A1], FWT [A0] -FWT [A1]) $

$ UFWT [A '] = merge (\ frac {FWT [A0'] + FWT [A1 ']} {2}, \ frac {FWT [A0'] - FWT [A1 ']} {2}) $

code:

#include <cstdio> 
#include <algorithm>        
#define N 19   
#define ll long long 
#define mod 998244353 
#define setIO(s) freopen(s".in","r",stdin)  
using namespace std;        
int qpow(int x,int y) 
{
    int tmp=1; 
    for(;y;y>>=1,x=(ll)x*x%mod)  
        if(y&1) tmp=(ll)tmp*x%mod; 
    return tmp;   
}
int lim,inv,A[1<<N],B[1<<N],C[1<<N],f[1<<N],g[1<<N];    
void FWT_or(int *a,int opt) 
{ 
    int i,j,k;    
    for(i=1;i<lim;i<<=1) 
    {
        for(j=0;j<lim;j+=i<<1)                                  
        {
            for(k=0;k<i;++k) 
            {
                if(opt==1) a[i+j+k]=1ll*(a[j+k]+a[i+j+k])%mod;   
                else a[i+j+k]=1ll*(a[i+j+k]+mod-a[j+k])%mod;     
                // a[j+k+i]=(ll)(1ll*a[j+k+i]+1ll*opt*a[j+k]+mod)%mod;      
            }
        }
    }
}           
void FWT_and(int *a,int opt) 
{     
    int i,j,k;  
    for(i=1;i<lim;i<<=1) 
    {
        for(j=0;j<lim;j+=i<<1) 
        {
            for(k=0;k<i;++k) 
            {
                (a[j+k]+=1ll*(opt*a[j+k+i]+mod)%mod)%=mod;   
            }
        }
    }
}     
void FWT_xor(int *a,int opt) 
{   
    int i,j,k;     
    for(i=1;i<lim;i<<=1) 
    {
        for(j=0;j<lim;j+=i<<1) 
        {
            for(k=0;k<i;++k) 
            {
                int tmp=a[j+k];    
                a[j+k]=(ll)(a[j+k]+a[j+k+i])%mod;   
                a[j+k+i]=(ll)(tmp-a[j+k+i]+mod)%mod;          
                if(opt==-1) 
                {
                    a[j+k]=(ll)inv*a[j+k]%mod;  
                    a[j+k+i]=(ll)inv*a[j+k+i]%mod;   
                }
            }
        }
    }
}
void test_or() 
{   
    int i,j; 
    for(i=0;i<lim;++i) f[i]=A[i],g[i]=B[i];  
    FWT_or(f,1),FWT_or(g,1);  
    for(i=0;i<lim;++i) f[i]=1ll*f[i]*g[i]%mod;              
    FWT_or(f,-1);  
    for(i=0;i<lim;++i) printf("%d ",f[i]);   
    printf("\n");   
}  
void test_and() 
{ 
    int i,j;  
    for(i=0;i<lim;++i) f[i]=A[i],g[i]=B[i];    
    FWT_and(f,1),FWT_and(g,1);  
    for(i=0;i<lim;++i) f[i]=1ll*f[i]*g[i]%mod;   
    FWT_and(f,-1);  
    for(i=0;i<lim;++i) printf("%d ",f[i]);  
    printf("\n");  
} 
void test_xor() 
{     
    int i,j;   
    for(i=0;i<lim;++i) f[i]=A[i],g[i]=B[i];   
    FWT_xor (f, 1), FWT_xor    
    for(i=0;i<lim;++i) f[i]=(ll)f[i]*g[i]%mod;   
    FWT_xor(f,-1);   
    for(i=0;i<lim;++i) printf("%d ",f[i]);   
    printf("\n");    
}
int main() 
{ 
    // setIO("input"); 
    int i,j,n; 
    scanf("%d",&n),lim=1<<n,inv=qpow(2,mod-2);                    
    for(i=0;i<lim;++i) scanf("%d",&A[i]),A[i]=(A[i]%mod+mod)%mod;   
    for(i=0;i<lim;++i) scanf("%d",&B[i]),B[i]=(B[i]%mod+mod)%mod; 
    test_or(),test_and(),test_xor();          
    return 0;  
}