解析

动态维护树的重心
关键性质：两棵树合并时，新的重心一定在两个原重心之间的路径上
把两个重心之间的路径提出来，利用splay本身的二叉结构二分即可
注意虚子树信息的处理
不要忘记pushdown

代码

#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define debug(...) fprintf(stderr,__VA_ARGS__)
const int N=3e5+100;
const int mod=1e9+7;
const double eps=1e-9;
inline ll read() {ll x(0),f(1);char c=getchar();while(!isdigit(c)){if(c=='-')f=-1;c=getchar();}while(isdigit(c)){x=(x<<1)+(x<<3)+c-'0';c=getchar();}return x*f;
}int n,m;
int flag;
int tr[N][2],f[N],rev[N];
int siz[N],siz0[N];
#define ls(u) tr[u][0]
#define rs(u) tr[u][1]
inline bool isroot(int x){return ls(f[x])!=x&&rs(f[x])!=x;
}
inline bool which(int x){return rs(f[x])==x;
}
inline void pushup(int x){if(x) siz[x]=siz[ls(x)]+siz[rs(x)]+siz0[x]+1;return;
}
inline void Rev(int x){if(x){rev[x]^=1;swap(tr[x][0],tr[x][1]);}return;
}
inline void pushdown(int x){if(rev[x]){rev[x]=0;Rev(tr[x][0]);Rev(tr[x][1]);}return;
}
void print(int x){if(!x) return;pushdown(x);printf("x=%d ls=%d rs=%d fa=%d siz=%d siz0=%d\n",x,ls(x),rs(x),f[x],siz[x],siz0[x]);print(ls(x));print(rs(x));
}
void Debug(){//printf("debug:\n");for(int i=1;i<=n;i++) if(isroot(i)) print(i);putchar('\n');
}
inline void rotate(int x){int fa=f[x],gfa=f[fa];int d=which(x),son=tr[x][d^1];f[x]=gfa;if(!isroot(fa)) tr[gfa][which(fa)]=x;f[fa]=x;tr[x][d^1]=fa;if(son){f[son]=fa;}tr[fa][d]=son;pushup(fa);pushup(x);return;
}
int zhan[N];
inline void splay(int x){int y=x,top(0);zhan[++top]=y;while(!isroot(y)) zhan[++top]=y=f[y];while(top){//printf("  pushdown:%d\n",y);pushdown(zhan[top--]);}for(int fa;fa=f[x],!isroot(x);rotate(x)){if(!isroot(fa)) which(x)==which(fa)?rotate(fa):rotate(x);}return;
}
inline void access(int x){for(int y=0;x;y=x,x=f[x]){splay(x);siz0[x]+=siz[rs(x)];siz0[x]-=siz[y];rs(x)=y;pushup(x);if(y) f[y]=x;}return;
}
inline void makeroot(int x){//if(flag) printf("makeroot:%d\n\n",x);access(x);//if(flag) printf("access:\n"),Debug();splay(x);Rev(x);return;
}
inline void link(int x,int y){makeroot(x);access(y);splay(y);siz0[y]+=siz[x];f[x]=y;pushup(y);return;
}
inline void split(int x,int y){//if(flag) printf("split:(%d %d)\n",x,y);makeroot(x);//printf("makeroot:\n");Debug();access(y);//printf("access:\n");Debug();splay(y);//printf("splay:\n");Debug();return;
}int fa[N];
int find(int x){return fa[x]==x?x:fa[x]=find(fa[x]);
}int solve(int x,int y){//printf("pre:\n");Debug();split(y,x);//Debug();int l(0),r(0),lsum(0),rsum(0),ans(0),tot=siz[x];while(x){pushdown(x);l=ls(x);r=rs(x);int lsiz=lsum+siz[l],rsiz=rsum+siz[r];//printf("x=%d l=%d r=%d tot=%d lsiz=%d rsiz=%d\n",x,l,r,tot,lsiz,rsiz);if(lsiz<=tot/2&&rsiz<=tot/2){			if(!ans||x<ans) ans=x;if(tot&1) break;}if(lsiz>rsiz){rsum+=siz[r]+siz0[x]+1;x=l;}else{lsum+=siz[l]+siz0[x]+1;x=r;}//printf("  lsum=%d rsum=%d\n",lsum,rsum);}splay(ans);return ans;
}int now;
char s[10];
int main() {
#ifndef ONLINE_JUDGEfreopen("a.in","r",stdin);freopen("a.out","w",stdout);
#endifn=read();m=read();for(int i=1;i<=n;i++){siz[i]=1;now^=i;fa[i]=i;}for(int i=1;i<=m;i++){scanf(" %s",s+1);//if(i==5) flag=1;if(s[1]=='X') printf("%d\n",now);else if(s[1]=='Q') printf("%d\n",find((int)read()));else{int x=read(),y=read();link(x,y);if(flag) printf("link:\n"),Debug();x=find(x);y=find(y);int z=solve(x,y);now^=x^y;now^=z;fa[z]=fa[x]=fa[y]=z;//printf("i=%d x=%d y=%d z=%d\n\n",i,x,y,z);}//if(flag) printf("finish:\n"),Debug();}return 0;
}