poj1679 The Unique MST(次小生成树)

题意：给你一个n个点m条边的无向图,问你该图的最小生成树是否唯一?如果唯一输出,树的权值,否则输出'Not Unique!'.

思路：本题求该无向图的次小生成树的权值是否等于最小生成树的权值，一个图的次小生成树的权值<=最小生成树的权值，至少有一条边与最小生成树不同，所以可以枚举最小生成树上的边，删除之后用剩下的边重新求最小生成树，求出新生成的最小生成树中权值最小就是为次小生成树的权值
Trick:删除一条边之后原图可能不连通，返回-1，注意考虑清楚

#include <cstdio>
#include <queue>
#include <cstring>
#include <iostream>
#include <cstdlib>
#include <algorithm>
#include <vector>
#include <map>
#include <string>
#include <set>
#include <ctime>
#include <cmath>
#include <cctype>
using namespace std;
#define maxn 110
const int maxm = 100*100+10;
#define LL long long
int cas=1,T;
struct Edge
{
int u,v,dist;
int id;           //边的编号
Edge(){}
Edge(int u,int v,int dist,int id):u(u),v(v),dist(dist),id(id){}
bool operator < (const Edge&rhs)const
{
return dist < rhs.dist;
}
};
int n,m;
Edge edges[maxm];
int pre[maxn];
vector <int> E;            //保存最小生成树的边的编号
int Find(int x)
{
return pre[x]==-1?x:pre[x]=Find(pre[x]);
}
void init()
{
m=0;
memset(pre,-1,sizeof(pre));
}
void AddEdge(int u,int v,int dist,int id)
{
edges[m++]=Edge(u,v,dist,id);
}
int Kruskal(int ID)
{
memset(pre,-1,sizeof(pre));
E.clear();
int sum = 0;
int cnt = 0;
sort(edges,edges+m);
for (int i = 0;i<m;i++)
{
if (edges[i].id == ID)
continue;
int u = edges[i].u;
int v = edges[i].v;
if (Find(u)!=Find(v))
{
E.push_back(edges[i].id);
pre[Find(u)]=Find(v);
sum+=edges[i].dist;
if (++cnt >=n-1)
break;
}
}
if (cnt < n-1)
return -1;
return sum;
}
int main()
{
//freopen("in","r",stdin);
scanf("%d",&T);
while (T--)
{
int mm;
scanf("%d%d",&n,&mm);
init();
for (int i = 0;i<mm;i++)
{
int u,v,d;
scanf("%d%d%d",&u,&v,&d);
AddEdge(u,v,d,i);
}
int ans1 = Kruskal(-1);
int ans2 = 1<<20;
vector <int>EE(E);
for (int i = 0;i<EE.size();i++)
{
int temp = Kruskal(EE[i]);
if (temp == -1)
continue;
ans2 = min(ans2,temp);
if (ans2 == ans1)
break;
}
if (ans1 == ans2)
puts("Not Unique!");
else
printf("%d\n",ans1);
}
//printf("time=%.3lf",(double)clock()/CLOCKS_PER_SEC);
return 0;
}

题目

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!