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!
```

The question ： Ask if the minimum spanning tree is unique .

analysis ： I want to make a small tree , Infer whether the second smallest spanning tree and the smallest spanning tree are equal .

The steps of finding the next smallest spanning tree ：

(1) First use Prime Find the minimum spanning tree MST, stay Prime Using a matrix at the same time mmax[ ][ ] Recorded in the MST Connect two random points in u,v Right in the only path of

The weight of the side with the largest value . practice ：Prime Is to add one node at a time t. Use this point to add MST The edge of is joined with its previous one MST Point of mmax Compare the values of .

(2) Enumerate edges other than the minimum spanning tree , And delete the edge with the largest weight on the ring where the edge is located .

(3) The tree with the smallest weight of all spanning trees obtained is the tree to be calculated .

The time complexity of the algorithm is O(n^2).

``` #include <iostream>
#include <cstring>
#include <cstdio>
using namespace std;
#define maxn 111
#define inf 0x3f3f3f3f
int map[maxn][maxn],mmax[maxn][maxn];//map Adjacency matrix is a graph ,mmax In the minimum spanning tree i To j The maximum edge weight of
bool used[maxn][maxn];// Determine whether the edge is added to the minimum spanning tree
int pre[maxn],dis[maxn];//pre be used for mmax The construction of , Put one in before loading MST The node of ,dis Used to build MST
void init(int n)
{
for (int i=;i<=n;i++)// Graph initialization
{
for (int j=;j<=n;j++)
{
if (i==j)
{
map[i][j]=;
}
else
{
map[i][j]=inf;
}
}
}
}
{
int u,v,w;
for (int i=;i<m;i++)// Read in the picture
{
scanf("%d%d%d",&u,&v,&w);
map[u][v]=map[v][u]=w;
}
}
int prime(int n)// structure MST
{
int ans=;
bool vis[maxn];
memset(vis,false,sizeof(vis));
memset(used,false,sizeof(used));
memset(mmax,,sizeof(mmax));
for (int i=;i<=n;i++)
{
dis[i]=map[][i];
pre[i]=;//1 Point for the first to put in MST The point of , Let's set it as the leading node of all points
}
pre[]=;
dis[]=;
vis[]=true;
for (int i=;i<=n;i++)
{
int min_dis=inf,k;
for (int j=;j<=n;j++)
{
if (vis[j]==&&min_dis>dis[j])
{
min_dis=dis[j];
k=j;
}
}
if (min_dis==inf)// If there is no minimum spanning tree
{
return -;
}
ans+=min_dis;
vis[k]=true;
used[k][pre[k]]=used[pre[k]][k]=true;// Mark as put in MST The point of
for (int j=;j<=n;j++)
{
if (vis[j])
{
mmax[j][k]=mmax[k][j]=max(mmax[j][pre[k]],dis[k]);// The largest edge of the smallest spanning tree ring
}
if (!vis[j]&&dis[j]>map[k][j])
{
dis[j]=map[k][j];
pre[j]=k;
}
}
}
return ans;// The sum of the weights of the minimum spanning tree
}
int smst(int n,int min_ans)//min_ans Is the sum of the weights of the minimum spanning tree
{
int ans=inf;
for (int i=;i<=n;i++)// Enumerate the edges outside the minimum spanning tree
{
for (int j=i+;j<=n;j++)
{
if (map[i][j]!=inf&&!used[i][j])
{
ans=min(ans,min_ans+map[i][j]-mmax[i][j]);// This side is the second smallest MST A weight of MST Add the edge and subtract the maximum of the ring where the edge is located MST edge
}
}
}
if (ans==inf)
{
return -;
}
return ans;
}
void solve(int n)
{
int ans=prime(n);
if (ans==-)
{
puts("Not Unique!");
return;
}
if (smst(n,ans)==ans)// Minor MST The weight is equal to MST explain MST Is not the only
{
printf("Not Unique!\n");
}
else
{
printf("%d\n",ans);
}
}
int main()
{
int t,n,m;
scanf("%d",&t);
while (t--)
{
scanf("%d%d",&n,&m);
init(n);
solve(n);
}
return ;
}```

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