「POJ1679 」The Unique MST

Posted 2021-02-09 yuns

题目描述

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‘.

输入格式

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.

输出格式

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

样例输入

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

样例输出e

3
Not Unique!

题目大意

• 给出一个有 n 个结点，m 条边的无向图，判断其最小生成树是否唯一。若唯一输出结果，不唯一输出“Not Unique!”

解题思路

• 我们考虑MST 什么时候唯一？显然当加入一条非树边时MST会形成环。
• 因此找到环上除了新加入的边外权值最大的边。当该边权值小于这条非树边，则MST唯一。
• 因此先利用Kruskal算法求出最小生成树结果，再将形成最小生成树的路径进行标记，每次去掉一个最小生成树的边，用其他边代替，若能找到最小生成树的结果还为原本结果，即说明最小生成树不唯一。