E - Minimum Spanning Tree Gym - 102220E (转化+贡献)
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)
.
Input
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
.
Output
For each test case, print a single line containing an integer, denoting the sum of all the edges' weight of MST(L(G))
.
Example
2
4
1 2 1
2 3 2
3 4 3
4
1 2 1
1 3 1
1 4 1
8
4
题解:题目给出一张图,让我们将每个边看成一个”点“,两“点”之间的权值为两边权之和。让我们找到这个“点”组成的图(题目命名为”线图“)的最小生成树的权值和。
我们可以从每条边权(即每个点的出边)的贡献入手,首先一个点的出边必须连通,否则构不成最小生成树。
那么对于特定的一个点,首先将其所有出边全部的权值加一遍,然后将其最小的一个边权乘以(这一点的度degree-2)即保证了最优解。(其实这样就是连degree-1条边使得保证最优解)
对于每一个点都这样,跑一遍即可。
#include<iostream>
#include<cstring>
#include<string>
#include<queue>
#include<stack>
#include<algorithm>
#include<stdio.h>
#include<map>
#include<set>
using namespace std;
typedef long long ll;
const int maxn=;
struct node
{
int v,w;
bool operator < (const node &r)const{
return w<r.w;
}
};
vector<node>G[maxn];
int main()
{
ios::sync_with_stdio();
int T;
cin>>T;
while(T--){
int n;
cin>>n;
for(int i=;i<=n;i++)G[i].clear();
for(int i=;i<=n-;i++){
int u,v,w;
cin>>u>>v>>w;
G[u].push_back((node){v,w});
G[v].push_back((node){u,w});
}
ll ans=;
for(int i=;i<=n;i++){
sort(G[i].begin(),G[i].end());
int minn=0x3f3f3f3f;
int degree=G[i].size();
for(int j=;j<degree;j++){
ans+=G[i][j].w;
minn=min(minn,G[i][j].w);
}
ans+=(ll)minn*(degree-);//当degree为1时与上面的ans相互消去
}
cout<<ans<<endl;
}
return ;
}
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