https://www.lydsy.com/JudgeOnline/problem.php?id=1977

题解：次小生成树

/*
*@Author:   STZG
*@Language: C++
*/
#include <bits/stdc++.h>
#include<iostream>
#include<algorithm>
#include<cstdlib>
#include<cstring>
#include<cstdio>
#include<string>
#include<vector>
#include<bitset>
#include<queue>
#include<deque>
#include<stack>
#include<cmath>
#include<list>
#include<map>
#include<set>
//#define DEBUG
#define RI register int
#define endl "\n"
using namespace std;
typedef long long ll;
//typedef __int128 lll;
const int N=300000+10;
const int M=100000+10;
const int MOD=1e9+7;
const double PI = acos(-1.0);
const double EXP = 1E-8;
const ll INF = 0x3f3f3f3f3f3f3f;
int t,n,m,k,p,l,r,u,v;
ll ans,cnt,flag,temp,sum;
int pre[N],dep[N],dp[N][22],maxl[N][22],maxll[N][22];
bool vis[N],used[N];
char str;

struct node{
    int u,v,w;
    node(){};
    node(int _u,int _v,int _w):u(_u),v(_v),w(_w){}
    bool operator <(const node&S)const{
        return w<S.w;
    }
}e[N];
vector<node>G[N];
void init(){
    memset(vis,0,sizeof(vis));
    for(int i=1;i<=n;i++){
        pre[i]=i;
    }
    sum=0;
    ans=INF;
}
int find(int x){return pre[x]==x?x:pre[x]=find(pre[x]);}
void marge(int x,int y){
    int fx=find(x);
    int fy=find(y);
    if(fx!=fy){
        pre[fy]=fx;
    }
}
void Kruskal(){
    sort(e+1,e+m+1);
    for(int i=1;i<=m&&cnt<n-1;i++){
        int fx=find(e[i].u);
        int fy=find(e[i].v);
        if(fx!=fy){
            cnt++;
            pre[fy]=fx;
            sum+=e[i].w;
            used[i]=1;
            G[e[i].u].push_back({e[i].u,e[i].v,e[i].w});
            G[e[i].v].push_back({e[i].v,e[i].u,e[i].w});
        }
    }
}
void dfs(int u){
    vis[u]=1;//cout<<G[u].size()<<endl;
    for(int i=0,j=G[u].size();i<j;i++){
        int v=G[u][i].v;
        if(!vis[v]){
            dep[v]=dep[u]+1;
            dp[v][0]=u;
            maxl[v][0]=G[u][i].w;
            maxll[v][0]=0;
            dfs(v);
        }
    }
}
void ST(){
    for(int i=0;i<21;i++){
        for(int j=1;j<=n;j++){
            dp[j][i+1]=dp[dp[j][i]][i];
            maxl[j][i+1]=max(maxl[j][i],maxl[dp[j][i]][i]);
            maxll[j][i+1]=(maxl[j][i]==maxl[dp[j][i]][i])?max(maxll[j][i],maxll[dp[j][i]][i]):min(maxl[j][i],maxl[dp[j][i]][i]);
        }
    }
}
int LCA(int x,int y,int w){
    int maxn[2]={0};
    if(dep[x]<dep[y]){
        swap(x,y);
    }
    while(dep[x]!=dep[y]){
        int i=log2(dep[x]-dep[y]);
        maxn[0]=max(maxn[0],maxl[x][i]);
        maxn[1]=max(maxn[1],maxll[x][i]);
        x=dp[x][i];
    }
    if(x==y){
        return maxn[0]==w?maxn[1]:maxn[0];
    }
    for(int i=log2(dep[x]);i>=0;i--){
        if(dp[x][i]!=dp[y][i]){
            maxn[0]=max(maxn[0],max(maxl[x][i],maxl[y][i]));
            maxn[1]=max(maxn[1],max(maxll[x][i],maxll[y][i]));
            x=dp[x][i];
            y=dp[y][i];
        }
    }
    if(maxl[x][0]>maxn[0]){
        if(maxll[x][0]>maxn[0])
            maxn[0]=maxl[x][0],maxn[1]=maxll[x][0];
        else maxn[0]=maxn[1],maxn[1]=maxl[x][0];
    }else if(maxl[x][0]<maxn[0]){
        if(maxl[x][0]>maxn[1])
            maxn[1]=maxl[x][0];
    }
    if(maxl[y][0]>maxn[0]){
        if(maxll[y][0]>maxn[0])
            maxn[0]=maxl[y][0],maxn[1]=maxll[y][0];
        else maxn[0]=maxn[1],maxn[1]=maxl[y][0];
    }else if(maxl[y][0]<maxn[0]){
        if(maxl[y][0]>maxn[1])
            maxn[1]=maxl[y][0];
    }//cout<<dp[x][0]<<endl;
    return maxn[0]==w?maxn[1]:maxn[0];
}
void TUMST(){
    for(int i=1;i<=m;i++){
        if(!used[i]){
            int tu=LCA(e[i].u,e[i].v,e[i].w);//cout<<sum<<endl;
            ans=min(ans,sum-tu+e[i].w);
        }
    }
}
int main()
{
#ifdef DEBUG
	freopen("input.in", "r", stdin);
	//freopen("output.out", "w", stdout);
#endif
    //ios::sync_with_stdio(false);
    //cin.tie(0);
    //cout.tie(0);
    //scanf("%d",&t);
    //while(t--){
    scanf("%d%d",&n,&m);
    init();
    for(int i=1;i<=m;i++){
        scanf("%d%d%d",&e[i].u,&e[i].v,&e[i].w);
    }
    Kruskal();
    dfs(1);
    ST();
    TUMST();
    cout<<ans<<endl;
    //}

#ifdef DEBUG
	printf("Time cost : %lf s\n",(double)clock()/CLOCKS_PER_SEC);
#endif
    //cout << "Hello world!" << endl;
    return 0;
}