P4717 模板快速沃尔什变换

Posted 2021-02-14 olinr

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给定长度为(2^n)两个序列(A,B)，设(C_i=sum_{joplus k}A_jB_k)
分别当(oplus)是or,and,xor时求出C

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第一行一个数n。
第二行(2^n)个数(A_0..A_{2^n-1})
第三行(2^n)个数(B_0..B_{2^n-1})

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三行每行(2^n)个数，分别代表(oplus)是or,and,xor时(C_0..C_{2^n-1})的值(mod 998244353)

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2
2 4 6 8
1 3 5 7

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2 22 46 250
88 64 112 56
100 92 68 60

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n≤17。

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背公式qwq

FWTor

[ A[i+l+k]+=A[i+k] ]

IFWTor

[ A[i+l+k]-=A[i+k] ]

FWTand

[ A[i+k]+=A[i+l+k] ]

IFWTand

[ A[i+k]-=A[i+l+k] ]

FWTxor

[ A[i+k]=A[i+k]+A[i+l+k] \ A[i+k+l]=A[i+k]-A[i+l+k] ]

IFWTxor

[ A[i+k]=frac{A[i+k]+A[i+l+k]}{2} \ A[i+k+l]=frac{A[i+k]-A[i+l+k]}{2} ]

#include<bits/stdc++.h>
#define LL long long
LL in() {
    char ch; LL x = 0, f = 1;
    while(!isdigit(ch = getchar()))(ch == '-') && (f = -f);
    for(x = ch ^ 48; isdigit(ch = getchar()); x = (x << 1) + (x << 3) + (ch ^ 48));
    return x * f;
}
const int maxn = 1e6 + 10;
const int mod = 998244353;
const int inv = 499122177;
int a[maxn], b[maxn], c[maxn], len;
void FWTor(int *A, int flag) {
    for(int l = 1; l < len; l <<= 1)
        for(int i = 0; i < len; i += (l << 1))
            for(int k = 0; k < l; k++)
                (A[i + l + k] += ((flag * A[i + k]) % mod + mod) % mod) %= mod;
}
void FWTand(int *A, int flag) {
    for(int l = 1; l < len; l <<= 1)
        for(int i = 0; i < len; i += (l << 1))
            for(int k = 0; k < l; k++)
                (A[i + k] += ((flag * A[i + l + k]) % mod + mod) % mod) %= mod;
}
void FWTxor(int *A, int flag) {
    for(int l = 1; l < len; l <<= 1)
        for(int i = 0; i < len; i += (l << 1))
            for(int k = 0; k < l; k++) {
                int tmp = A[i + k];
                A[i + k] = 1LL * (tmp + A[i + l + k]) % mod * inv % mod * flag % mod;
                A[i + l + k] = 1LL * ((tmp - A[i + l + k]) % mod + mod) % mod * inv % mod * flag % mod;
            }
}
void workor() {
    FWTor(a, 1), FWTor(b, 1);
    for(int i = 0; i < len; i++) c[i] = 1LL * a[i] * b[i] % mod;
    FWTor(a, -1), FWTor(b, -1), FWTor(c, -1);
    for(int i = 0; i < len; i++) printf("%d%c", c[i], i + 1 == len? '
' : ' ');
}

void workand() {
    FWTand(a, 1), FWTand(b, 1);
    for(int i = 0; i < len; i++) c[i] = 1LL * a[i] * b[i] % mod;
    FWTand(a, -1), FWTand(b, -1), FWTand(c, -1);
    for(int i = 0; i < len; i++) printf("%d%c", c[i], i + 1 == len? '
' : ' ');
}
void workxor() {
    FWTxor(a, 2), FWTxor(b, 2);
    for(int i = 0; i < len; i++) c[i] = 1LL * a[i] * b[i] % mod;
    FWTxor(a, 1), FWTxor(b, 1), FWTxor(c, 1);
    for(int i = 0; i < len; i++) printf("%d%c", c[i], i + 1 == len? '
' : ' ');
}
int main() {
    len = 1 << in();
    for(int i = 0; i < len; i++) a[i] = in();
    for(int i = 0; i < len; i++) b[i] = in();
    workor();
    workand();
    workxor();
    return 0;
}