[思维]Minimum Spanning Tree

Posted 2020-12-02 lllxq

题目描述

In the mathematical discipline of graph theory, the line graph of a simple undirected weighted graph G is another simple undirected weighted graph L(G) that represents the adjacency between every two edges in G.

Precisely speaking, for an undirected weighted graph G without loops or multiple edges, its line graph L(G) is a graph such that:
·Each vertex of L(G) represents an edge of G.
·Two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint in G, and the weight of such edge between this two vertices is the sum of their corresponding edges‘ weight.

A minimum spanning tree(MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible.

Given a tree G, please write a program to find the minimum spanning tree of L(G).

输入

The first line of the input contains an integer T(1≤T≤1000), denoting the number of test cases.

In each test case, there is one integer n(2≤n≤100000) in the first line, denoting the number of vertices of G.

For the next n−1 lines, each line contains three integers u,v,w(1≤u,v≤n,u≠v,1≤w≤109), denoting a bidirectional edge between vertex u and v with weight w.

It is guaranteed that ∑n≤106.

输出

For each test case, print a single line containing an integer, denoting the sum of all the edges‘ weight of MST(L(G)).

样例输入 Copy

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

样例输出 Copy

8
4
题意：将原图中的边看作新图中的点，原图中的点看作新图中的边，且新图中的边权为它所连接的两个点的权值之和（即原图中的两边权之和），求新图的最小生成树。
思路：考虑对一个点来说，与其相连的所边要构成联通，其最小花费是什么？即找到权值最小的边，其他所有边都与它相连。
因为对于每一个点都要单独考虑，所以单独计算每一个点的该最小花费，求和即为答案。