HDU-6057 Kanade's convolution

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题面

Description

Give you two arrays (A[0..2^m-1]) and (B[0..2^m-1]).

Please calculate array (C[0..2^m-1])
[ C[k]=sum_{i~and~j=k}A[i~xor~j]*B[i~or~j] ]
You just need to print (sum_{i=0}^{2^m-1}C[i]*1526^i ~mod~998244353)

(m<=19)

(0leq A[i],B[i]< 998244353)

Input

There is only one test case.

The first line consists of one integer (m).

The second line consists of (2^m) integers (A[0..2^m-1])

The third line consists of (2^m) integers (B[0..2^m-1])

Output

Output only one integer means the answer.

Sample Input

2
1 2 3 4
5 6 7 8

Sample Output

568535691

题意

[ C[k]=sum_{i~and~j=k}A[i~xor~j]*B[i~or~j] ]

计算上式

题解

我在做这题的时候没太有思路,去搜了题解,有一篇题解感觉说的很好

https://www.cnblogs.com/Tiw-Air-OAO/p/10274225.html

这种题,推式子显然不是我这种非专业数学选手掌握的,那么怎样提高自己做出这种题目的概率呢?

打表大概是一个很好的办法,这篇题解里也是这样入手的

~ (1) (2) (3) (4)
i 0 0 1 1
j 0 1 0 1
i xor j 0 1 1 0
i or j 0 1 1 1
i and j 0 0 0 1

我们令(a = i~xor~ j,b=i~or~j, c=i~and~j)

我们可以发现,显然,当异或值某一位为1的时候,或该位上必定为1,此时,i和j是该位上是可以有两种情况的,01或者10,而异或值为0,或值为1,和异或值为1,或值为0,i和j都是唯一确定的,也就是说我们可以根据异或值按位1的数量来判断i和j能够产生a和b的(i,j)数目,也就是(2^{bit(a)}),其中bit(a)表示a中1的数量

这样,我们一开始的时候,就把a(x)乘上(2^{bit(a)}),这样我们就可以不考虑原来的i和j而直接考虑a,b了

仔细观察,(a oplus b = c)

但是这个是有条件的,上面也说了,异或某一位是1,或该位上必定是1,也就是说,(a~and~b=a),这个条件我们不好控制,但我们可以用之前的bit数组来控制,(a~and~b=a Leftrightarrow bit(b)-bit(a) = bit(c))

我们可以把a数组中bit(i)=k的数放在(A[bit[i]][i]),b同理,然后做fwt得到c[i-j],然后只保留c中(C[bit[i]][i])的数即可。

代码

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll mod = 998244353;
const ll inv2 = (mod + 1) >> 1;

void fwt_xor(ll a[], int len, int op) {
    for (int h = 2; h <= len; h <<= 1) {
        for (int j = 0; j < len; j += h) {
            for (int k = j; k < j + h / 2; k++) {
                ll u = a[k], t = a[k + h / 2];
                a[k] = (u + t) % mod;
                a[k + h / 2] = (u - t + mod) % mod;
                if (op == -1) {
                    a[k] = a[k] * inv2 % mod;
                    a[k + h / 2] = a[k + h / 2] * inv2 % mod;
                }
            }
        }
    }
}

const ll N = 1 << 20;
ll a[N], b[N];
int bit[N];
ll A[21][N], B[21][N];
ll C[21][N];
int main() {
    int m;
    scanf("%d", &m);
    int n = (1 << m);
    for (int i = 0; i < n; i++) {
        scanf("%lld", &a[i]);
    }
    for (int i = 0; i < n; i++) {
        scanf("%lld", &b[i]);
    }
    for (int i = 0; i < n; i++) {
        bit[i] = 0;
        int x = i;
        while (x) {
            if (x & 1) bit[i]++;
            x >>= 1;
        }
    }
    for (int i = 0; i < n; i++) {
        A[bit[i]][i] = a[i] * (1ll << bit[i]) % mod;
        B[bit[i]][i] = b[i];
    }
    for (int i = 0; i <= m; i++) {
        fwt_xor(A[i], n, 1);
        fwt_xor(B[i], n, 1);
    }
    for (int i = 0; i <= m; i++) {
        for (int j = 0; j <= i; j++) {
            for (int k = 0; k < n; k++) {
                C[i - j][k] = (C[i - j][k] + A[j][k] * B[i][k] % mod) % mod;
            }
        }
    }
    for (int i = 0; i <= m; i++) {
        fwt_xor(C[i], n, -1);
    }
    ll now = 1;
    ll ans = 0;
    for (int i = 0; i < n; i++) {
        ans = (ans + C[bit[i]][i] * now % mod) % mod;
        now = now * 1526 % mod;
    }
    printf("%lld
", ans);
    return 0;
}

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