【BZOJ1016】【Luogu P4208】 [JSOI2008]最小生成树计数 最小生成树，矩阵树定理

蛮不错的一道题，遗憾就遗憾在数据范围会导致暴力轻松跑过。

最小生成树的两个性质：

• 不同的最小生成树，相同权值使用的边数一定相同。

• 不同的最小生成树，将其都去掉同一个权值的所有边，其连通性一致。

这样我们随便跑一个\(MST\)，就可以知道所有\(MST\)边的构造情况。由于性质二，我们可以考虑枚举每一种权值的所有边，保留所有非此权值的树边，看可以连出来多少种不同的最小生成树。也就是按照权值构造最小生成树，这个过程满足乘法原理。

#include <bits/stdc++.h>
using namespace std;

#define int long long
const int N = 100 + 5;
const int M = 1000 + 5;
const int Mod = 31011;

struct Len {
	int u, v, w;

	bool operator < (Len rhs) const {
		return w < rhs.w;
	}
}l[M];

vector <Len> v;

int n, m, S[N];

int find (int x) {
	return S[x] == x ? x : S[x] = find (S[x]);
}

vector <int> val, use, tot;

vector <int> :: iterator it;

void kruskal () {
	sort (l + 1, l + 1 + m);
	for (int i = 0; i <= n; ++i) S[i] = i;
	for (int i = 1; i <= m; ++i) {
		int fu = find (l[i].u);
		int fv = find (l[i].v);
		it = lower_bound (val.begin (), val.end (), l[i].w);
		if (it == val.end ()) {
			val.push_back (l[i].w);
			use.push_back (0);
			tot.push_back (1);
		} else {
			tot[it - val.begin ()]++;
		}
		if (fu != fv) {
			S[fu] = fv;
			it = lower_bound (val.begin (), val.end (), l[i].w);
			use[it - val.begin ()]++;
			v.push_back (l[i]);
		}
	}
}

int mat[N][N];

int gauss (int n) {
	int ret = 1;
	for (int i = 1; i <= n; ++i) {
		for (int k = i + 1; k <= n; ++k) {
			while (mat[k][i]) {
				int d = mat[i][i] / mat[k][i];
				for (int j = i; j <= n; ++j) {
					(((mat[i][j] -= d * mat[k][j]) %= Mod) += Mod) %= Mod;
				}
				swap (mat[i], mat[k]); ret = -ret;
			}
		}
		(((ret *= mat[i][i]) %= Mod) += Mod) %= Mod;
	}
	return abs (ret);
}

void add_edge (int u, int v) {
	mat[u][u]++;
	mat[v][v]++;
	mat[u][v]--;
	mat[v][u]--;
}

int sep[N]; 

int solve () {
	kruskal ();
	if (v.size () < n - 1) return 0;
	int ans = 1;
	for (int i = 0; i < val.size (); ++i) {
		memset (mat, 0, sizeof (mat));
		if (use[i] == 0 || tot[i] == use[i]) continue;
		for (int j = 0; j <= n; ++j) S[j] = j;
		for (int j = 0; j < v.size (); ++j) {
			if (v[j].w != val[i]) {
				S[find (v[j].u)] = find (v[j].v);
			}
		}
		int cnt = 0;
		for (int i = 1; i <= n; ++i) {
			sep[++cnt] = find (i);
		}
		sort (sep + 1, sep + 1 + cnt);
		cnt = unique (sep + 1, sep + 1 + cnt) - sep - 1;
		for (int j = 1; j <= m; ++j) {
			if (l[j].w == val[i]) {
				int fu = find (l[j].u);
				int fv = find (l[j].v);
				fu = lower_bound (sep + 1, sep + 1 + cnt, fu) - sep;
				fv = lower_bound (sep + 1, sep + 1 + cnt, fv) - sep;
				add_edge (fu, fv);
			}
		}
		(ans *= gauss (use[i])) %= Mod;
	}
	return ans;
}

signed main () {
//	freopen ("data.in", "r", stdin);
	cin >> n >> m;
	for (int i = 1; i <= m; ++i) {
		static int u, v, w;
		cin >> u >> v >> w;
		l[i] = (Len) {u, v, w};
	}
	cout << solve () << endl;
}