BZOJ 4671:异或图题解

主要思路

我们发现,直接算一个连通块非常的困难。这个时候可以尝试容斥。我们发现至少一个的特别好算:划分至少\(i\)个连通块的答案记为\(g(i)\)。\(g(i)\)比较好算,可以考虑 DFS 枚举出每一个点所在的连通块根,然后把连通情况放入线性基中,如果线性基插入失败,那么说明当前的情况不能被表示出,且该状态符合划分,那么计入小答案\(tot\),最后对答案的贡献就是:\(2^{tot}\)。

现在考虑如何用\(g(i)\)算恰好\(i\)个连通块的答案\(f(i)\)。考虑搞下容斥系数,使得\(f(i)\)只被计算一次:

\[ \sum_{i = 1}^n \begin{Bmatrix} n \\ i \end{Bmatrix} coefficient(i) = [n = 1] \]

打表出来就是:

\[ coefficient(i) = (-1)^{i – 1} (i – 1)! \]

代码

// BZ4671.cpp
#pragma GCC optimize(3)
#include <bits/stdc++.h>
#define ll long long

using namespace std;

const int MAX_N = 110, MAX_S = 65;

ll fac[MAX_N], linear_bas[61], answer = 0, stat[MAX_N], g[MAX_N];
int S, n, len, mp[MAX_S][MAX_N][MAX_N], aff[MAX_N], hbit;
char str[1005];
bool insert(ll x)
{
    for (ll i = hbit - 1; i >= 0; i--)
        if ((x >> i) & 1)
        {
            if (linear_bas[i] == 0)
            {
                linear_bas[i] = x;
                return true;
            }
            x ^= linear_bas[i];
        }
    return false;
}

void dfs(int x, int org)
{
    if (x == n + 1)
    {
        hbit = 0, memset(linear_bas, 0, sizeof(linear_bas));
        for (int curt_g = 1; curt_g <= S; curt_g++)
        {
            stat[curt_g] = hbit = 0;
            for (int i = 1; i <= n; i++)
                for (int j = i + 1; j <= n; j++)
                    if (aff[i] != aff[j])
                        stat[curt_g] |= (1LL * mp[curt_g][i][j] << hbit), hbit++;
        }
        int tot = 0;
        for (int curt_g = 1; curt_g <= S; curt_g++)
            if (!insert(stat[curt_g]))
                tot++;
        g[org] += (1LL << tot);
        return;
    }
    for (int i = 1; i <= org; i++)
        aff[x] = i, dfs(x + 1, org);
    aff[x] = org + 1, dfs(x + 1, org + 1);
}

int main()
{
    scanf("%lld", &S);
    for (int i = 1; i <= S; i++)
    {
        scanf("%s", str + 1), len = strlen(str + 1);
        if (n == 0)
            n = (1 + (int)(sqrt(8 * len + 1))) >> 1;
        int cur = 1;
        for (int x = 1; x <= n; x++)
            for (int y = x + 1; y <= n; y++)
                mp[i][x][y] = (str[cur++] == '1');
    }

    for (int i = fac[0] = 1; i <= n; i++)
        fac[i] = 1LL * i * fac[i - 1];

    dfs(1, 0);
    for (int i = 1; i <= n; i++)
        answer += 1LL * ((i & 1) ? 1LL : -1LL) * fac[i - 1] * g[i];
    printf("%lld", answer);
    return 0;
}

 

发布者

Kal0rona

江西师大附中全机房最弱

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