# poj The Unique MST 次小生成樹（入門級）

The Unique MST
 Time Limit: 1000MS Memory Limit: 10000K Total Submissions: 30039 Accepted: 10754

Description

Given a connected undirected graph, tell if its minimum spanning tree is unique.

Definition 1 (Spanning Tree): Consider a connected, undirected graph G = (V, E). A spanning tree of G is a subgraph of G, say T = (V’, E’), with the following properties:
1. V’ = V.
2. T is connected and acyclic.

Definition 2 (Minimum Spanning Tree): Consider an edge-weighted, connected, undirected graph G = (V, E). The minimum spanning tree T = (V, E’) of G is the spanning tree that has the smallest total cost. The total cost of T means the sum of the weights on all
the edges in E’.

Input

The first line contains a single integer t (1 <= t <= 20), the number of test cases. Each case represents a graph. It begins with a line containing two integers n and m (1 <= n <= 100), the number of
nodes and edges. Each of the following m lines contains a triple (xi, yi, wi), indicating that xi and yi are connected by an edge with weight = wi. For any two nodes, there is at most one edge connecting them.

Output

For each input, if the MST is unique, print the total cost of it, or otherwise print the string ‘Not Unique!’.

Sample Input

```2
3 3
1 2 1
2 3 2
3 1 3
4 4
1 2 2
2 3 2
3 4 2
4 1 2
```

Sample Output

```3
Not Unique!
```

Source

POJ Monthly–2004.06.27 [email protected]

AC程式碼：

``````#include<iostream>
#include<cstdio>
#include<cstring>
using namespace std;
#define MAX 105
const int INF=0x3f3f3f3f;
int g[MAX][MAX],dist[MAX],mmax[MAX][MAX];  ///g儲存地圖  dist儲存從起點到其餘各點的距離 maxn儲存從i到j的最大邊權值
int pre[MAX]; ///pre儲存j的離它最近的是哪個點
bool mark[MAX]; ///相當於vis  用來標記該點是否已經用過
bool connect[MAX][MAX]; ///儲存i-j的那條邊是否加入了最小生成樹  false 加入 true  沒有
int mst,mint; ///mst儲存最小生成樹的權值和
int n,m;
int prim()
{
int res=0,fa,p,min,i,j;
memset(mmax,0,sizeof(mmax));
for(i=1;i<=n;i  )
{
dist[i]=g[1][i];
pre[i]=1;
mark[i]=false;
}
dist[1]=0;
mark[1]=true;
for(i=1;i<n;i  )
{
p=-1;min=INF;
for(j=1;j<=n;j  )
{
if(!mark[j]&&dist[j]<min)
{
p=j;
min=dist[j];
}
}
if(p==-1) return res;
mark[p]=true;
res =dist[p];
fa=pre[p]; ///找到離p最近的點
connect[fa][p]=false;
connect[p][fa]=false;
mmax[fa][p]=min;
///遍歷所有的點 求其餘點到p的最大權值
for(j=1;j<=n;j  )
mmax[j][p]=(mmax[fa][p]>mmax[j][fa])?mmax[fa][p]:mmax[j][fa];
for(j=1;j<=n;j  )
{
if(!mark[j]&&dist[j]>g[p][j])
{
dist[j]=g[p][j];
pre[j]=p;
}
}
}
return res;
}
int main()
{
int tc;
scanf("%d",&tc);
while(tc--)
{
scanf("%d %d",&n,&m);
memset(g,INF,sizeof(g));
memset(connect,false,sizeof(connect));
while(m--)
{
int u,v,c;
scanf("%d %d %d",&u,&v,&c);
g[u][v]=c;
g[v][u]=c;
connect[u][v]=true;
connect[v][u]=true;
}
mst=prim();
int i,j;
bool flag=false;
for(i=1;i<=n;i  )
for(j=1;j<=n;j  )
{
///如果i-j這條邊加入了最小生成樹 或者i-j這條路不通  continue
if(connect[i][j]==false||g[i][j]==INF)
continue;
///如果加入的邊和刪除的邊的大小是一樣的  說明次小生成樹的權值和等於最小生成樹的權值和
///也就是說最小生成樹不唯一
if(g[i][j]==mmax[i][j])
{
flag=true;
break;
}
}
if(flag)
printf("Not Unique!\n");
else
printf("%d\n",mst);
}
return 0;
}``````