CF1137F
题意
Lena在玩火柴的时候突然好奇用火柴烧树会怎么样。
由于Lena很善良不想烧树，所以她决定在脑子中模拟烧树的过程。
我们认为烧树的过程是这样的：
树上每个点有个独一无二的优先级（初始第i个节点为i），每次会烧掉优先级最小的点。
现在有q个操作，每个操作形如： 1. 把u号点的优先级改成当前最大的优先级+1 2. 询问如果把这棵树烧掉，u号点会被第几个烧掉。 3. 询问如果把这棵树烧掉，u和v哪个点会先被烧掉。
\(n,q\le 2\cdot 10^5\)
题解
显然3操作可以转为两个2操作。
设树的根为当前优先级最大的点，\(w_i\)为\(i\)子树中优先级的\(\max\)
对于\(2\)操作，那么\(i\)在\(u\)之前烧掉当且仅当\(w_i<w_u\)或\(w_i=w_u\)且\(u\)是\(i\)祖先。
对于\(1\)操作，设原来的根是\(v\)，优先级为\(w\)，则相当于把\(u\dots v\)的路径上所有点\(i\)（不包括\(u\)）\(w_i=w\)，然后\(w_u=w+1\)，把根换成\(u\)。相当于，把\(u\)到根路径\(w_i=w\)，然后把根换成\(u\)，然后把\(u\)到根路径（只有\(u\)一个点）\(w_i=w+1\)
那么就可以直接LCT+BIT了。
代码
/*
Author: QAQ-Automaton
LANG: C++
PROG: F.cpp
Mail: cnyalilk@vip.qq.com
*/
#include<bits/stdc++.h>
#define debug(...) fprintf(stderr,__VA_ARGS__)
#define DEBUG printf("Passing [%s] in LINE %d\n",__FUNCTION__,__LINE__)
#define Debug debug("Passing [%s] in LINE %d\n",__FUNCTION__,__LINE__)
#define all(x) x.begin(),x.end()
#define x first
#define y second
using namespace std;
typedef long long ll;
typedef pair<int,int> pii;
#define inf 0x3f3f3f3f3f3f3f3f3f3f3f3f3f3f3f3f
const double eps=1e-8;
const double pi=acos(-1.0);
template<class T>bool chkmin(T &a,T b){return a>b?a=b,1:0;}
template<class T>bool chkmax(T &a,T b){return a<b?a=b,1:0;}
template<class T>T sqr(T a){return a*a;}
template<class T>T mmin(T a,T b){return a<b?a:b;}
template<class T>T mmax(T a,T b){return a>b?a:b;}
template<class T>T aabs(T a){return a<0?-a:a;}
template<class T>bool dcmp(T a,T b){return a>b;}
template<int *a>bool cmp_a(bool x,bool y){return a[x]<a[y];}
#define min mmin
#define max mmax
#define abs aabs
namespace io {
const int SIZE = (1 << 21) + 1;
char ibuf[SIZE], *iS, *iT, obuf[SIZE], *oS = obuf, *oT = oS + SIZE - 1, c, qu[55]; int f, qr;
// getchar
#define gc() (iS == iT ? (iT = (iS = ibuf) + fread (ibuf, 1, SIZE, stdin), (iS == iT ? EOF : *iS ++)) : *iS ++)
// print the remaining part
inline void flush () {
fwrite (obuf, 1, oS - obuf, stdout);
oS = obuf;
}
// putchar
inline void putc (char x) {
*oS ++ = x;
if (oS == oT) flush ();
}
// input a signed integer
inline bool read (signed &x) {
for (f = 1, c = gc(); c < '0' || c > '9'; c = gc()) if (c == '-') f = -1;else if(c==EOF)return 0;
for (x = 0; c <= '9' && c >= '0'; c = gc()) x = x * 10 + (c & 15); x *= f;
return 1;
}
inline bool read (long long &x) {
for (f = 1, c = gc(); c < '0' || c > '9'; c = gc()) if (c == '-') f = -1;else if(c==EOF)return 0;
for (x = 0; c <= '9' && c >= '0'; c = gc()) x = x * 10 + (c & 15); x *= f;
return 1;
}
inline bool read (char &x) {
x=gc();
return x!=EOF;
}
inline bool read(char *x){
while((*x=gc())=='\n' || *x==' '||*x=='\r')if(*x==EOF)return 0;
while(!(*x=='\n'||*x==' '||*x=='\r'))*(++x)=gc();
*x=0;
return 1;
}
template<typename A,typename ...B>
inline bool read(A &x,B &...y){
return read(x)&&read(y...);
}
// print a signed integer
inline bool write (signed x) {
if (!x) putc ('0'); if (x < 0) putc ('-'), x = -x;
while (x) qu[++ qr] = x % 10 + '0', x /= 10;
while (qr) putc (qu[qr --]);
return 0;
}
inline bool write (long long x) {
if (!x) putc ('0'); if (x < 0) putc ('-'), x = -x;
while (x) qu[++ qr] = x % 10 + '0', x /= 10;
while (qr) putc (qu[qr --]);
return 0;
}
inline bool write (char x) {
putc(x);
return 0;
}
inline bool write(const char *x){
while(*x){putc(*x);++x;}
return 0;
}
inline bool write(char *x){
while(*x){putc(*x);++x;}
return 0;
}
template<typename A,typename ...B>
inline bool write(A x,B ...y){
return write(x)||write(y...);
}
//no need to call flush at the end manually!
struct Flusher_ {~Flusher_(){flush();}}io_flusher_;
}
using io :: read;
using io :: putc;
using io :: write;
namespace bit{
int a[400005];
int n=400000,y;
void add(int x,int y){if(x)for(;x<=n;x+=x&-x)a[x]+=y;}
int sum(int x){for(y=0;x;x^=x&-x)y+=a[x];return y;}
}
namespace lct{
int fa[200005],son[200005][2],t;
int siz[200005],rev[200005],col[200005];
void push_up(int x){siz[x]=1+siz[son[x][0]]+siz[son[x][1]];}
void put_rev(int x){rev[x]^=1;swap(son[x][0],son[x][1]);}
void rotate(int x){
int f=fa[x],g=fa[f],slr=son[f][1]==x;
fa[f]=x;fa[x]=g;
if(son[g][1]==f)son[g][1]=x;
if(son[g][0]==f)son[g][0]=x;
fa[son[f][slr]=son[x][slr^1]]=f;
son[x][slr^1]=f;
push_up(f);
push_up(x);
}
#define ntr(x) (fa[x] && (son[fa[x]][1]==x||son[fa[x]][0]==x))
void push_tag(int x){
if(ntr(x))push_tag(fa[x]);
if(rev[x]){put_rev(son[x][0]);put_rev(son[x][1]);rev[x]=0;}
col[son[x][0]]=col[x];
col[son[x][1]]=col[x];
}
void splay(int x){
push_tag(x);
while(ntr(x)){
int f=fa[x];
if(ntr(f)){
if((son[f][1]==x)==(son[fa[f]][1]==f))rotate(f);
else rotate(x);
}
rotate(x);
}
}
void access(int x,int c=0){
int y=0;
for(;x;y=x,x=fa[x]){
splay(x);
son[x][1]=0;
push_up(x);
bit::add(col[x],-siz[x]);
son[x][1]=y;
push_up(x);
}
col[y]=c;
bit::add(c,siz[y]);
}
void mtr(int x){
access(x,t);
splay(x);
put_rev(x);
access(x,++t);
}
void link(int x,int y){
mtr(x);
fa[x]=y;
}
int ask(int x){
splay(x);
return bit::sum(col[x])-siz[son[x][0]];
}
}
char s[15];
signed main(){
#ifdef QAQAutoMaton
freopen("F.in","r",stdin);
freopen("F.out","w",stdout);
#endif
int n,q,u,v;
read(n,q);
for(int i=1;i<n;++i){
read(u,v);
lct::link(u,v);
}
lct::t=0;
// for(int i=1;i<=n;++i){bit::add(lct::col[i],-1);lct::col[i]=0;lct::rev[i]=0;}
for(int i=1;i<=n;++i)lct::mtr(i);
for(;q;--q){
read(s);
read(u);
if(*s=='u'){lct::mtr(u);}
else if(*s=='w'){write(lct::ask(u),'\n');}
else{read(v); write(lct::ask(u)<lct::ask(v)?u:v,'\n');}
}
return 0;
}