FWT整理

FWT

FWT

by AmanoKumiko

1.Or

\[A_i=∑_{j|i=i}Aj \]

\[FWT[A]=merge(FWT[A_0],FWT[A_0]+FWT[A_1]) \]

\[UFWT[A]=merge(UFWT[A_0],UFWT[A_1]-UFWT[A_0]) \]

---

2.And

\[A_i=∑_{j\ and\ i=i}Aj \]

\[FWT[A]=merge(FWT[A_0]+FWT[A_1],FWT[A_1]) \]

\[UFWT[A]=merge(UFWT[A_0]-UFWT[A_1],UFWT[A_1]) \]

---

3.Xor

\[i\oplus j=popcount(i\ and \ j)\ mod\ 2 \]

\[A_i=∑_{i\oplus j=0}A_j-∑_{i\oplus j=1}A_j \]

\[FWT[A]=merge(FWT[A_0]+FWT[A_1],FWT[A_0]-FWT[A_1]) \]

\[UFWT[A]=merge(\frac{UFWT[A_0]+UFWT[A_1]}{2},\frac{UFWT[A_0]-UFWT[A_1]}{2}) \]

---

4.code

struct poly{
	int len;
	LL val[N];
	void Or(int opt){
		for(int mid=2;mid<len;mid<<=1){
			for(int i=0;i<len;i+=mid){
				F(j,i,i+mid/2)val[j+mid/2]+=val[j]*opt;
			}
		}
	}
	void And(int opt){
		for(int mid=2;mid<len;mid<<=1){
			for(int i=0;i<len;i+=mid){
				F(j,i,i+mid/2)val[j]+=val[j+mid/2]*opt;
			}
		}
	}
	void Xor(int opt){
		for(int mid=2;mid<len;mid<<=1){
			for(int i=0;i<len;i+=mid){
				F(j,i,i+mid/2){
					LL l=val[j],r=val[j+mid/2];
					val[j]=(l+r)/(opt==1?1:2);val[j+mid/2]=(l-r)/(opt==1?1:2);
				}
			}
		}
	}
}