【HDU 4408】Minimum Spanning Tree（最小生成树计数）

Problem Description

XXX is very interested in algorithm. After learning the Prim algorithm and Kruskal algorithm of minimum spanning tree, XXX finds that there might be multiple solutions. Given an undirected weighted graph with n (1<=n<=100) vertexes and m (0<=m<=1000) edges, he wants to know the number of minimum spanning trees in the graph.

 

Input

There are no more than 15 cases. The input ends by 0 0 0.
For each case, the first line begins with three integers --- the above mentioned n, m, and p. The meaning of p will be explained later. Each the following m lines contains three integers u, v, w (1<=w<=10), which describes that there is an edge weighted w between vertex u and vertex v( all vertex are numbered for 1 to n) . It is guaranteed that there are no multiple edges and no loops in the graph.

 

Output

For each test case, output a single integer in one line representing the number of different minimum spanning trees in the graph.
The answer may be quite large. You just need to calculate the remainder of the answer when divided by p (1<=p<=1000000000). p is above mentioned, appears in the first line of each test case.

 

Sample Input

5 10 12

2 5 3

2 4 2

3 1 3

3 4 2

1 2 3

5 4 3

5 1 3

4 1 1

5 3 3

3 2 3

0 0 0

 

Sample Output

4

 

Source

2012 ACM/ICPC Asia Regional Jinhua Online

 

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题意是给定n个点，m条边的无向图，求最小生成树的个数对p取模。

用kruscal计算最小生成树时，每次取连接了两个不同联通块的最小的边。也就是先处理d₁条c₁长度的边，再处理d₂条c₂长度的边。长度相同的边无论怎么选，最大联通情况都是固定的。
分别对c_i长度的边产生的几个联通块计算生成树数量再乘起来，然后把这些联通块缩点，再计算c_i+1长度的边。

生成树计数用Matrix-Tree定理，上一篇是无重边的，这题的缩点后是会产生重边的，Matrix-tree也适用：

Kirchhoff矩阵任意n-1阶子矩阵的行列式的绝对值就是无向图的生成树的数量。

Kirchhoff矩阵的定义是度数矩阵-邻接矩阵。

1、G的度数矩阵D[G]：n*n的矩阵，D_ii等于V_i的度数，其余为0。
2、G的邻接矩阵A[G]：n*n的矩阵， V_i、V_j之间有边直接相连，则 A_ij=ij之间的边数，否则为0。

并查集fa[i]是当前长度之前，节点所属的联通块，ka[i]是当前长度的边连接后它在的联通块。

#include <cstdio>
#include <cstring>
#include <algorithm>
#include <vector>
using namespace std;
typedef long long ll;
const int N=;
const int M=;
ll n,m,p,ans;
vector<int>gra[N];
struct edge{
    int u,v,w;
}e[M];
int cmp(edge a,edge b){
    return a.w<b.w;
}
ll mat[N][N],g[N][N];
ll fa[N],ka[N],vis[N];
ll det(ll c[][N],ll n){
    ll i,j,k,t,ret=;
    for(i=;i<n;i++)
    for(j=;j<n;j++) c[i][j]%=p;
    for(i=; i<n; i++){
        for(j=i+; j<n; j++)
            while(c[j][i]){
                t=c[i][i]/c[j][i];
                for(k=i; k<n; k++)
                    c[i][k]=(c[i][k]-c[j][k]*t)%p;
                swap(c[i],c[j]);
                ret=-ret;
            }
        if(c[i][i]==)
            return 0L;
        ret=ret*c[i][i]%p;
    }
    return (ret+p)%p;
}
ll find(ll a,ll f[]){
    return f[a]==a?a:find(f[a],f);
}
void matrix_tree(){//对当前长度的边连接的每个联通块计算生成树个数
    for(int i=;i<n;i++)if(vis[i]){//当前长度的边连接了i节点
        gra[find(i,ka)].push_back(i);//将i节点压入所属的联通块
        vis[i]=;//一边清空vis数组
    }
    for(int i=;i<n;i++)
    if(gra[i].size()>){//联通块的点数为1时生成树数量是1
        memset(mat,,sizeof mat);//清空矩阵
        int len=gra[i].size();
        for(int j=;j<len;j++)
        for(int k=j+;k<len;k++){//构造这个联通块的矩阵（有重边）
            int u=gra[i][j],v=gra[i][k];
            if(g[u][v]){
                mat[k][j]=(mat[j][k]-=g[u][v]);
                mat[k][k]+=g[u][v];mat[j][j]+=g[u][v];
            }
        }
        ans=ans*det(mat,gra[i].size()-)%p;
        for(int j=;j<len;j++)fa[gra[i][j]]=i;//缩点
    }
    for(int i=;i<n;i++)
    {
        gra[i].clear();
        ka[i]=fa[i]=find(i,fa);
    }
}
int main(){
    while(scanf("%lld%lld%lld",&n,&m,&p),n){
        for(int i=;i<m;i++){
            int u,v,w;
            scanf("%d%d%d",&u,&v,&w);
            u--;v--;
            e[i]=(edge){u,v,w};
        }
        sort(e,e+m,cmp);
        memset(g,,sizeof g);
        ans=;
        for(ll i=;i<n;i++)ka[i]=fa[i]=i;
        for(ll i=;i<=m;i++){//边从小到大加入
            if(i&&e[i].w!=e[i-].w||i==m)//处理完长度为e[i-1].w的所有边
                matrix_tree();//计算生成树
            ll u=find(e[i].u,fa),v=find(e[i].v,fa);//连的两个缩点后的点
            if(u!=v)//如果不是一个
            {
                vis[v]=vis[u]=;
                ka[find(u,ka)]=find(v,ka);//两个分量在一个联通块里。
                g[u][v]++,g[v][u]++;//邻接矩阵
            }
        }
        int flag=;
        for(int i=;i<n;i++)if(fa[i]!=fa[i-])flag=;
        printf("%lld\n",flag?ans%p:);//注意p可能为1，这样m=0时如果ans不%p就会输出1
    }
}