luoguP4336 [SHOI2016]黑暗前的幻想乡 容斥原理 + 矩阵树定理

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自然地想到容斥原理

然后套个矩阵树就行了

求行列式的时候只有换行要改变符号啊QAQ

复杂度为\(O(2^n * n^3)\)

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#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>
using namespace std;

#define ri register int
#define rep(io, st, ed) for(ri io = st; io <= ed; io ++)
#define drep(io, ed, st) for(ri io = ed; io >= st; io --)

const int sid = 20;
const int mod = 1e9 + 7;
inline void inc(int &a, int b) { a += b; if(a >= mod) a -= mod; }
inline void dec(int &a, int b) { a -= b; if(a < 0) a += mod; }
inline int mul(int a, int b) { return 1ll * a * b % mod; }
inline int inv(int a) {
	int ret = 1;
	for(int k = mod - 2; k; k >>= 1, a = mul(a, a))
		if(k & 1) ret = mul(ret, a);
	return ret;
}
	
int N, ans;
int M[sid], G[sid][sid];
struct edge { int u, v; } E[sid][400];

inline int Guass(int n) {
	int sign = 1;
	rep(i, 1, n) {
		int pos = i;
		rep(j, i + 1, n) if(G[j][i]) pos = j;
		swap(G[i], G[pos]); 
		if(i != pos) sign *= -1;
		if(!G[i][i]) return 0;
		int Inv = inv(G[i][i]);
		rep(j, i + 1, n) {
			int t = mul(G[j][i], Inv);
			rep(k, i, n) dec(G[j][k], mul(G[i][k], t));
		}
	}
	int ret = 1;
	rep(i, 1, n) ret = mul(ret, G[i][i]);
	if(sign == 1) return ret;
	else return mod - ret;
}

inline int calc(int S) {
	memset(G, 0, sizeof(G));
	rep(i, 1, N - 1) {
		if(!(S & (1 << i - 1))) continue;
		rep(j, 1, M[i]) {
			int u = E[i][j].u, v = E[i][j].v;
			inc(G[u][u], 1); inc(G[v][v], 1);
			dec(G[u][v], 1); dec(G[v][u], 1);
		}
	}
	return Guass(N - 1);
}

inline void dfs(int o, int S, int num) {
	if(o == N) {
		if((N - 1 - num) & 1) dec(ans, calc(S));
		else inc(ans, calc(S));
		return;
	}
	dfs(o + 1, S, num);
	dfs(o + 1, S | (1 << o - 1), num + 1);
}
	
int main() {
	freopen("pp.in", "r", stdin);
	cin >> N;
	rep(i, 1, N - 1) {
		cin >> M[i];
		rep(j, 1, M[i]) 
		cin >> E[i][j].u >> E[i][j].v;
	}
	dfs(1, 0, 0);
	printf("%d\n", ans);
	return 0;
}