BZOJ3510 首都（LCT）

　　即动态维护树的重心。考虑合并后的新重心一定在两棵树的重心的连线上。于是对每个点维护其子树大小，合并时在这条链的splay上二分即可。至于如何维护子树大小，见https://blog.csdn.net/neither_nor/article/details/52979425。明明都看那么多题解说要注意pushdown，结果还是因为这个调了一年，心态爆炸。（下面代码复杂度可能是假的，因为dfs完没有splay叶子，反正也没人卡我就不管了）

#include<iostream>
#include<cstdio>
#include<cmath>
#include<cstdlib>
#include<cstring>
#include<algorithm>
using namespace std;
#define ll long long
#define N 100010
#define M 200010
#define lson tree[k].ch[0]
#define rson tree[k].ch[1]
#define lself tree[tree[k].fa].ch[0]
#define rself tree[tree[k].fa].ch[1]
char getc(){char c=getchar();while (c<'A'||c>'Z') c=getchar();return c;}
int gcd(int n,int m){return m==?n:gcd(m,n%m);}
int read()
{
    int x=,f=;char c=getchar();
    while (c<''||c>'') {if (c=='-') f=-;c=getchar();}
    while (c>=''&&c<='') x=(x<<)+(x<<)+(c^),c=getchar();
    return x*f;
}
int n,m,ans;
struct data{int ch[],fa,rev,size_f,size;
}tree[N];
void up(int k){tree[k].size=tree[lson].size+tree[rson].size+tree[k].size_f+;}
void rev(int k){if (k) swap(lson,rson),tree[k].rev^=;}
void down(int k){if (tree[k].rev) rev(lson),rev(rson),tree[k].rev=;}
bool isroot(int k){return lself!=k&&rself!=k;}
int whichson(int k){return rself==k;}
void push(int k){if (!isroot(k)) push(tree[k].fa);down(k);}
void move(int k)
{
    int fa=tree[k].fa,gf=tree[fa].fa,p=whichson(k);
    if (!isroot(fa)) tree[gf].ch[whichson(fa)]=k;tree[k].fa=gf;
    tree[tree[k].ch[!p]].fa=fa,tree[fa].ch[p]=tree[k].ch[!p];
    tree[k].ch[!p]=fa,tree[fa].fa=k;
    up(fa),up(k);
}
void splay(int k)
{
    push(k);
    while (!isroot(k))
    {
        int fa=tree[k].fa;
        if (!isroot(fa))
            if (whichson(k)^whichson(fa)) move(k);
            else move(fa);
        move(k);
    }
}
void access(int k)
{
    for (int t=;k;t=k,k=tree[k].fa)
    {
        splay(k);
        tree[k].size_f+=tree[rson].size;
        rson=t;
        tree[k].size_f-=tree[rson].size;
        up(k);
    }
}
void makeroot(int k){access(k),splay(k),rev(k);}
int findroot(int k){access(k),splay(k);for (;lson;k=lson) down(k);splay(k);return k;}
int dfs(int k,int s,int r)
{
    if (!k) return n+;
    down(k);
    if (s-(tree[k].size-tree[lson].size+r)<=(s>>))
    {
        int x=dfs(rson,s,r);
        if (x<=n) return x;else return k;
    }
    else return dfs(lson,s,r+tree[k].size-tree[lson].size);
}
int find(int k){down(k);if (rson) return find(rson);else return k;}
void link(int x,int y)
{
    int p=findroot(x),q=findroot(y);ans^=p,ans^=q;
    makeroot(y),access(x),splay(x),tree[y].fa=x;
    tree[x].size_f+=tree[y].size,up(x);
    makeroot(p),access(q),splay(q);int s=tree[q].size;
    p=dfs(q,s,),splay(p);
    if (tree[p].size-tree[tree[p].ch[]].size<=(s>>))
    splay(q=find(tree[p].ch[])),p=min(p,q);
    makeroot(p);ans^=p;
}
int main()
{
#ifndef ONLINE_JUDGE
    freopen("bzoj3510.in","r",stdin);
    freopen("bzoj3510.out","w",stdout);
    const char LL[]="%I64d\n";
#else
    const char LL[]="%lld\n";
#endif
    n=read(),m=read();
    for (int i=;i<=n;i++) ans^=i,tree[i].size=;
    while (m--)
    {
        char c=getc();
        if (c=='X') printf("%d\n",ans);
        if (c=='A') link(read(),read());
        if (c=='Q') printf("%d\n",findroot(read()));
    }
    return ;
}