#line 2 "graph_tree/lca.hpp"
#include<vector>
#include<cmath>
#include<tuple>
#line 4 "graph_tree/graph_template.hpp"
#include<iostream>
/**
* @brief グラフテンプレート
*/
using graph=std::vector<std::vector<int>>;
template<typename T>
using graph_w=std::vector<std::vector<std::pair<int,T>>>;
graph load_graph(int n,int m){graph g(n);for(int i=0;i<m;++i){int s,t;std::cin>>s>>t;--s;--t;g[s].push_back(t);g[t].push_back(s);}return g;}
graph load_digraph(int n,int m){graph g(n);for(int i=0;i<m;++i){int s,t;std::cin>>s>>t;--s;--t;g[s].push_back(t);}return g;}
graph load_graph0(int n,int m){graph g(n);for(int i=0;i<m;++i){int s,t;std::cin>>s>>t;g[s].push_back(t);g[t].push_back(s);}return g;}
graph load_digraph0(int n,int m){graph g(n);for(int i=0;i<m;++i){int s,t;std::cin>>s>>t;g[s].push_back(t);}return g;}
graph load_tree(int n){graph g(n);for(int i=0;i<n-1;++i){int s,t;std::cin>>s>>t;--s;--t;g[s].push_back(t);g[t].push_back(s);}return g;}
graph load_tree0(int n){graph g(n);for(int i=0;i<n-1;++i){int s,t;std::cin>>s>>t;g[s].push_back(t);g[t].push_back(s);}return g;}
graph load_treep(int n){graph g(n);for(int i=0;i<n-1;++i){int t;std::cin>>t;g[i+1].push_back(t);g[t].push_back(i+1);}return g;}
template<typename T>graph_w<T> load_graph_weight(int n,int m){graph_w<T> g(n);for(int i=0;i<m;++i){int s,t;T u;std::cin>>s>>t>>u;--s;--t;g[s].emplace_back(t,u);g[t].emplace_back(s,u);}return g;}
template<typename T>graph_w<T> load_digraph_weight(int n,int m){graph_w<T> g(n);for(int i=0;i<m;++i){int s,t;T u;std::cin>>s>>t>>u;--s;--t;g[s].emplace_back(t,u);}return g;}
template<typename T>graph_w<T> load_graph0_weight(int n,int m){graph_w<T> g(n);for(int i=0;i<m;++i){int s,t;T u;std::cin>>s>>t>>u;g[s].emplace_back(t,u);g[t].emplace_back(s,u);}return g;}
template<typename T>graph_w<T> load_digraph0_weight(int n,int m){graph_w<T> g(n);for(int i=0;i<m;++i){int s,t;T u;std::cin>>s>>t>>u;g[s].emplace_back(t,u);}return g;}
template<typename T>graph_w<T> load_tree_weight(int n){graph_w<T> g(n);for(int i=0;i<n-1;++i){int s,t;T u;std::cin>>s>>t>>u;--s;--t;g[s].emplace_back(t,u);g[t].emplace_back(s,u);}return g;}
template<typename T>graph_w<T> load_tree0_weight(int n){graph_w<T> g(n);for(int i=0;i<n-1;++i){int s,t;T u;std::cin>>s>>t>>u;g[s].emplace_back(t,u);g[t].emplace_back(s,u);}return g;}
template<typename T>graph_w<T> load_treep_weight(int n){graph_w<T> g(n);for(int i=0;i<n-1;++i){int t;T u;std::cin>>t>>u;g[i+1].emplace_back(t,u);g[t].emplace_back(i+1,u);}return g;}
#line 5 "graph_tree/depth.hpp"
/**
* @brief 根からの深さ
*/
std::vector<int> depth(const graph& g,int start){
std::vector<int>memo(g.size());
auto f=[&](auto f,int v,int p)->int{
int mx=0;
for(auto t:g[v]){
if(t==p)continue;
mx=std::max(mx,f(f,t,v));
}
return memo[v]=mx+1;
};
f(f,start,-1);
return memo;
}
#line 1 "graph_tree/distance.hpp"
std::vector<int> distance(const graph& G,int start){
std::vector<int>memo(G.size());
auto f=[&](auto f,int v,int p,int i)->void{
for(auto t:G[v]){
if(t==p)continue;
f(f,t,v,i+1);
}
memo[v]=i;
};
f(f,start,-1,0);
return memo;
}
#line 2 "data_structure/arg_rmq.hpp"
#include<assert.h>
#line 4 "data_structure/arg_rmq.hpp"
#include<stack>
#include<numeric>
#line 7 "data_structure/arg_rmq.hpp"
#include<algorithm>
#line 3 "data_structure/small_rmq.hpp"
/**
* RMQ(small) &lt;O(N),O(1)&gt;(N<=64)
*/
template<typename T>
class small_rmq{
using u64=unsigned long long;
std::vector<u64>table;
std::vector<T> v;
public:
small_rmq(std::vector<T> v):v(v){
assert(v.size()<=64);
std::vector<int>tmp(v.size());
table.resize(v.size(),0);
std::stack<T>stk;
for(int i=0;i<(int)v.size();++i){
tmp.resize(v.size());
while(!stk.empty()&&v[stk.top()]>=v[i]){
stk.pop();
}
tmp[i]=stk.empty()?-1:stk.top();
stk.emplace(i);
}
for(int i=0;i<(int)v.size();++i){
if(tmp[i]!=-1)table[i]=table[tmp[i]]|(1ULL<<(tmp[i]));
}
}
T query(int l,int r){
assert(l!=r);
const u64 tmp=table[r-1]&~((1ULL<<l)-1);
if(tmp==0)return r-1;
else return __builtin_ctzll(tmp);
}
};
#line 3 "data_structure/sparse_table.hpp"
#include<functional>
#line 2 "alga/maybe.hpp"
#include<cassert>
/**
* @brief Maybe
* @see https://ja.wikipedia.org/wiki/%E3%83%A2%E3%83%8A%E3%83%89_(%E3%83%97%E3%83%AD%E3%82%B0%E3%83%A9%E3%83%9F%E3%83%B3%E3%82%B0)#Maybe%E3%83%A2%E3%83%8A%E3%83%89
*/
template<typename T>
struct maybe{
bool _is_none;
T val;
maybe():_is_none(true){}
maybe(T val):_is_none(false),val(val){}
T unwrap()const{
assert(!_is_none);
return val;
}
T unwrap_or(T e)const{
return _is_none?e:val;
}
bool is_none()const{return _is_none;}
bool is_some()const{return !_is_none;}
};
template<typename T,typename F>
auto expand(F op){
return [&op](const maybe<T>& s,const maybe<T>& t){
if(s.is_none())return t;
if(t.is_none())return s;
return maybe<T>(op(s.unwrap(),t.unwrap()));
};
}
#line 7 "data_structure/sparse_table.hpp"
/**
* @brief SparseTable
*/
template<typename T,typename F>
class sparse_table{
F f;
std::vector<std::vector<T>>data;
public:
sparse_table(std::vector<T> v,F f=F()):f(f){
int n=v.size(),log=log2(n)+1;
data.resize(n,std::vector<T>(log));
for(int i=0;i<n;i++)data[i][0]=v[i];
for(int j=1;j<log;j++)for(int i=0;i+(1<<(j-1))<n;i++){
data[i][j]=f(data[i][j-1],data[i+(1<<(j-1))][j-1]);
}
}
maybe<T> get(int l,int r){
if(l==r)return maybe<T>();
if(r<l)std::swap(l,r);
int k=std::log2(r-l);
return maybe<T>(f(data[l][k],data[r-(1<<k)][k]));
}
};
#line 3 "functional/argmin.hpp"
/**
* @brief 最小値とその位置
*/
template<typename T,typename E>
struct argmin{
std::pair<T,E> operator()(const std::pair<T,E>& s,const std::pair<T,E>& t){
return s.second<=t.second?s:t;
}
};
#line 11 "data_structure/arg_rmq.hpp"
/**
* @brief RangeArgminQuery &lt;O(N),O(1)&gt;
* @see https://noshi91.hatenablog.com/entry/2018/08/16/125415
*/
template<typename T>
class arg_rmq{
constexpr static int b=64;
std::vector<T>v;
std::vector<small_rmq<T>*>backet;
sparse_table<std::pair<int,T>,argmin<int,T>>* st=0;
public:
arg_rmq(std::vector<T>v):v(v){
std::vector<std::pair<int,T>>tmp2;
for(int i=0;i<(int)v.size();i+=b){
std::vector<T>tmp;
T mn=std::numeric_limits<T>::max();
int idx=-1;
for(int j=0;i+j<(int)v.size()&&j<b;j++){
tmp.push_back(v[i+j]);
if(mn>v[i+j]){
mn=v[i+j];
idx=i+j;
}
}
tmp2.emplace_back(idx,mn);
backet.push_back(new small_rmq<T>(tmp));
}
st=new sparse_table<std::pair<int,T>,argmin<int,T>>(tmp2);
}
maybe<std::pair<int,T>> query(int s,int t){
if(s==t)return maybe<std::pair<int,T>>();
if(s/b==t/b){
int idx=s/b*b+backet[s/b]->query(s%b,t%b);
return maybe<std::pair<int,T>>(std::make_pair(idx,v[idx]));
}
std::pair<int,T> res=std::make_pair(-1,std::numeric_limits<T>::max());
{
int idx=s/b*b+backet[s/b]->query(s%b,b);
res=argmin<int,T>()(res,std::make_pair(idx,v[idx]));
}
if(s/b+1!=t/b)res=argmin<int,T>()(res,st->get(s/b+1,t/b).unwrap());
if(t%b!=0){
int idx=t/b*b+backet[t/b]->query(0,t%b);
res=argmin<int,T>()(res,std::make_pair(idx,v[idx]));
}
return maybe<std::pair<int,T>>(res);
}
};
#line 9 "graph_tree/lca.hpp"
/**
* @brief LCA &lt;O(N),O(1)&gt;(HL分解と同等の速さ)
*/
class lca{
std::vector<int>data;
std::vector<int>comp_data;
std::vector<int>start;
arg_rmq<int>*st;
std::vector<int> __dist;
public:
lca(){}
lca(std::vector<std::vector<int>>v,int s){
data.resize(v.size()*2-1);
comp_data.resize(v.size()*2-1);
start.resize(v.size());
int i=0;
__dist=distance(v,s);
auto f=[&](auto f,int n,int p)->void{
start[n]=i;
data[i]=n;
comp_data[i++]=__dist[n];
for(int t:v[n]){
if(t==p)continue;
f(f,t,n);
data[i]=n;
comp_data[i++]=__dist[n];
}
};
f(f,s,-1);
st=new arg_rmq<int>(comp_data);
}
int query(int p,int q){
return data[st->query(std::min(start[p],start[q]),std::max(start[p],start[q])+1).unwrap().first];
}
int dist(int p,int q){
return __dist[p]+__dist[q]-2*__dist[query(p,q)];
}
};
LCA <O(N),O(1)>(HL分解と同等の速さ)