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Sol

\(LCT\)动态维护树重心

方法一

因为只有加边，所以可以暴力启发式合并，维护重心

维护子树信息，子树大小不超过一半

复杂度两只\(log\)

方法二

扣出两个重心的链，链上二分找

每次\(Splay\)重心，应该是一只\(log\)的吧。。。

# include <bits/stdc++.h>
# define RG register
# define IL inline
# define Fill(a, b) memset(a, b, sizeof(a))
using namespace std;
typedef long long ll;
const int _(1e5 + 5);

IL int Input(){
	RG int x = 0, z = 1; RG char c = getchar();
	for(; c < '0' || c > '9'; c = getchar()) z = c == '-' ? -1 : 1;
	for(; c >= '0' && c <= '9'; c = getchar()) x = (x << 1) + (x << 3) + (c ^ 48);
	return x * z;
}

int n, m, Xor, fa[_], ch[2][_], rev[_], S[_], sum[_], val[_], rt[_];
char op;

IL int Son(RG int x){
	return ch[1][fa[x]] == x;
}

IL int Isroot(RG int x){
	return ch[0][fa[x]] != x && ch[1][fa[x]] != x;
}

IL void Reverse(RG int x){
	if(!x) return;
	rev[x] ^= 1, swap(ch[0][x], ch[1][x]);
}

IL void Pushdown(RG int x){
	if(!rev[x]) return;
	Reverse(ch[0][x]), Reverse(ch[1][x]), rev[x] ^= 1;
}

IL void Update(RG int x){
	sum[x] = sum[ch[0][x]] + sum[ch[1][x]] + val[x] + 1;
}

IL void Rotate(RG int x){
	RG int y = fa[x], z = fa[y], c = Son(x);
	if(!Isroot(y)) ch[Son(y)][z] = x; fa[x] = z;
	ch[c][y] = ch[!c][x], fa[ch[c][y]] = y;
	ch[!c][x] = y, fa[y] = x, Update(y);
}

IL void Splay(RG int x){
	S[S[0] = 1] = x;
	for(RG int y = x; !Isroot(y); y = fa[y]) S[++S[0]] = fa[y];
	while(S[0]) Pushdown(S[S[0]--]);
	for(RG int y = fa[x]; !Isroot(x); Rotate(x), y = fa[x])
		if(!Isroot(y)) Son(x) ^ Son(y) ? Rotate(x) : Rotate(y);
	Update(x);
}

IL void Access(RG int x){
	for(RG int y = 0; x; y = x, x = fa[x]){
		Splay(x), val[x] += sum[ch[1][x]] - sum[y];
		ch[1][x] = y, Update(x);
	}
}

IL void Makeroot(RG int x){
	Access(x), Splay(x), Reverse(x);
}

IL void Split(RG int x, RG int y){
	Makeroot(x), Access(y), Splay(y);
}

IL void Adjust(RG int x, RG int y){
	Split(x, y);
	RG int flg = 0, t = y, g = 0, size = sum[y] >> 1, sl = 0, sr = 0;
	while(t){
		Pushdown(t);
		RG int ls = ch[0][t], rs = ch[1][t], ssl = sum[ls] + sl, ssr = sum[rs] + sr;
		if(ssl <= size && ssr <= size){
			if(!flg || t < g) g = t, flg = 1;
			if(ssl == ssr) break;
		}
		if(ssr > ssl) sl += sum[ls] + val[t] + 1, t = rs;
		else sr += sum[rs] + val[t] + 1, t = ls;
	}
	Splay(g);
	Xor ^= x ^ y ^ g, rt[x] = rt[y] = rt[g] = g;
}

IL int Findrt(RG int x){
	return x == rt[x] ? x : rt[x] = Findrt(rt[x]);
}

IL void Link(RG int x, RG int y){
	Makeroot(x), Makeroot(y);
	fa[x] = y, val[y] += sum[x], Update(y);
	Adjust(Findrt(x), Findrt(y));
}

int main(RG int argc, RG char *argv[]){
	n = Input(), m = Input();
	for(RG int i = 1; i <= n; ++i) rt[i] = i, Xor ^= i, sum[i] = 1;
	for(RG int i = 1, x, y; i <= m; ++i){
		scanf(" %c", &op);
		if(op == 'A') x = Input(), y = Input(), Link(x, y);
		else if(op == 'Q') x = Input(), printf("%d\n", Findrt(x));
		else scanf(" %*s"), printf("%d\n", Xor);
	}
	return 0;
}