[NOI2014]魔法森林（LCT）

蒟蒻又发现自己还没写过LCT……

首先显然按照权值a从小到大排序，维护b的最小生成树。然后直接扫，代价是加入b的最大值，然后动态加边，因为有边权，所以在lct中边应该理解为点。每次连接(u,v)时，若不连通则直接连起来，反之则需选择b最大的边断开。

#include<bits/stdc++.h>
using namespace std;
const int N=2e5+7;
struct node{int a,b,x,y;}q[N];
int n,m,len,fa[N],lc[N],rc[N],rev[N],b[N],val[N],s[N],f[N];
bool cmp(node a,node b){return a.y<b.y;}
int dir(int x){return x==rc[fa[x]];}
int find(int x){return x==f[x]?x:f[x]=find(f[x]);}
void union1(int x,int y){x=find(x),y=find(y);if(x!=y)f[x]=y;}
bool isroot(int x){return!fa[x]||lc[fa[x]]!=x&&rc[fa[x]]!=x;}
void pushup(int x)
{
    s[x]=x;
    if(lc[x]&&val[s[lc[x]]]>val[s[x]])s[x]=s[lc[x]];
    if(rc[x]&&val[s[rc[x]]]>val[s[x]])s[x]=s[rc[x]];
}
void pushdown(int x)
{
    if(rev[x])
    {
        swap(lc[x],rc[x]);
        if(lc[x])rev[lc[x]]^=1;
        if(rc[x])rev[rc[x]]^=1;
        rev[x]=0;
    }
}
void rotate(int x)
{
    int y=fa[x],z=fa[y],w=x==lc[y]?rc[x]:lc[x];
    if(z&&!isroot(y))(y==lc[z]?lc[z]:rc[z])=x;
    fa[x]=z,fa[y]=x;
    if(w)fa[w]=y;
    if(x==lc[y])rc[x]=y,lc[y]=w;else lc[x]=y,rc[y]=w;
    pushup(y),pushup(x);
}
void splay(int x)
{
    int len=1,y=x;b[1]=x;
    while(!isroot(y))b[++len]=fa[y],y=fa[y];
    for(int i=len;i;i--)pushdown(b[i]);
    while(!isroot(x))
    {
        if(!isroot(fa[x]))
        {
            if(dir(x)==dir(fa[x]))rotate(fa[x]);
            else rotate(x);
        }
        rotate(x);
    }
    pushup(x);
}
void access(int x)
{
    int y=0;
    while(x)
    {
        splay(x),rc[x]=y;
        if(y)fa[y]=x;
        pushup(x),y=x,x=fa[x];
    }
}
int findrt(int x)
{
    access(x),splay(x),pushdown(x);
    while(lc[x])x=lc[x];
    splay(x);
    return x;
}
void reverse(int x){access(x),splay(x),rev[x]^=1;}
void link(int x,int y){reverse(x),fa[x]=y;}
void cut(int x,int y){reverse(x),access(y),splay(y),lc[y]=fa[x]=0,pushup(y);}
int query(int x,int y){reverse(x),access(y),splay(y);return s[y];}
int main()
{
    scanf("%d%d",&n,&m);
    for(int i=1;i<=m;i++)scanf("%d%d%d%d",&q[i].a,&q[i].b,&q[i].x,&q[i].y);
    sort(q+1,q+m+1,cmp);
    for(int i=1;i<=n+m;i++)f[i]=s[i]=i;
    for(int i=n+1;i<=n+m;i++)val[i]=q[i-n].x;
    int ans=2e9;
    for(int i=1;i<=m;i++)
    {
        int u=q[i].a,v=q[i].b;
        bool flag=1;
        if(find(u)==find(v))
        {
            int w=query(u,v);
            if(val[w]>q[i].x)cut(q[w-n].a,w),cut(w,q[w-n].b);else flag=0;
        }
        else union1(u,v);
        if(flag)link(u,i+n),link(i+n,v);
        if(find(1)==find(n))ans=min(ans,q[i].y+val[query(1,n)]);
    }
    if(ans<2e9)printf("%d",ans);else puts("-1");
}