Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices, whose i-th edge connects Vertex u_i and Vertex v_i.
Vertex i has an integer a_i written on it.
For every integer k from 1 through N, solve the following problem:

We will make a sequence by lining up the integers written on the vertices along the shortest path from Vertex 1 to Vertex k, in the order they appear. Find the length of the longest increasing subsequence of this sequence.

Here, the longest increasing subsequence of a sequence A of length L is the subsequence A_{i_1} , A_{i_2} , ... , A_{i_M} with the greatest possible value of M such that 1 \leq i_1 < i_2 < ... < i_M \leq L and A_{i_1} < A_{i_2} < ... < A_{i_M}.

Constraints

2 \leq N \leq 2 \times 10^5

1 \leq a_i \leq 10^9

1 \leq u_i , v_i \leq N

u_i \neq v_i

The given graph is a tree.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 ... a_N
u_1 v_1
u_2 v_2
:
u_{N-1} v_{N-1}

Output

Print N lines. The k-th line, print the length of the longest increasing subsequence of the sequence obtained from the shortest path from Vertex 1 to Vertex k.

Sample Input 1

10
1 2 5 3 4 6 7 3 2 4
1 2
2 3
3 4
4 5
3 6
6 7
1 8
8 9
9 10

Sample Output 1

1
2
3
3
4
4
5
2
2
3

For example, the sequence A obtained from the shortest path from Vertex 1 to Vertex 5 is 1,2,5,3,4. Its longest increasing subsequence is A_1, A_2, A_4, A_5, with the length of 4.

Solution in C++ (GCC 9.2.1) (judge verdict: Runtime Error)

#include <bits/stdc++.h>
#define int long long
#define fastio ios_base::sync_with_stdio(false);cin.tie(NULL);cout.tie(NULL);
#define td(v) v.begin(),v.end()
#define tdr(v) v.rbegin(),v.rend()
#define endl "\n"
#define Matrix vector<vector<int> >
using namespace std;
const int MOD = 1e9+7;
const int MOD2 = 1e9 + 9;
const int INF = 5e18;
 
int gcd(int a, int b){return (b == 0 ? a : gcd(b, a%b));}
int lcm(int a,int b){return (a*b)/gcd(a,b);}
 
 
inline long long mod(long long n, long long m){
    long long ret = n%m;
    if(ret < 0) ret += m;
    return ret;
}
 
int exp(int a, int b){
    int result = 1;
    while (b > 0){
        if (b & 1)
            result = result*a;
 
        b >>= 1;
        a = a*a;
    }
    return result;
}
vector<int>adj[234567];
vector<int>val(234567);
vector<bool>flag(234567,false);
vector<int>vet;
vector<int>res;
void dfs(int pos){
    flag[pos]=true;
    int x = val[pos];
    int guard;
    bool incrementou = false;
    auto it = lower_bound(td(vet),x);
    if(it==vet.end()){
        incrementou=true;
        vet.push_back(x);
    }
    else{
        guard=*it;
        *it=x;
    }
    res[pos]=vet.size();
    for(int i=0;i<adj[pos].size();i++){
        if(!flag[adj[pos][i]])
            dfs(adj[pos][i]);
    }
    if(incrementou){
        vet.pop_back();
    }
    else{
        *it=guard;
    }
}
void solve(){
    int n;
    cin>>n;
    res.assign(n+1,0);
    for(int i=1;i<=n;i++){
        cin>>val[i];
    }
    for(int i=0;i<n-1;i++){
        int x,y; cin>>x>>y;
        adj[x].push_back(y);
        adj[y].push_back(x);
    }
    dfs(1);
    for(int i=1;i<=n;i++){
        cout<<res[i]<<endl;
    }
}
main(){
    fastio; 
    int x=0;
    int t=1;
    // cin>>t;
    while(t--)
        solve();
}