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: Time Limit Exceeded)

#include<iostream>
#include<vector>
#include<algorithm>
#define MAXN 200005
using namespace std;

int n;
int a[MAXN];
vector<int> edge[MAXN];
int ans[MAXN];
vector<int> lis;
int dp[MAXN];

void dfs(int i,vector<int> lis,int dp[],int maxn){
    lis.push_back(a[i]);
    int len = lis.size();
    dp[len-1] = 1;
    for(int j=0;j<len;++j){
        if(lis[j]<lis[len-1]) dp[len-1] = max(dp[len-1],dp[j]+1);
    }
    maxn = max(maxn,dp[len-1]);
    ans[i] = maxn;
    for(int j=0;j<(int)edge[i].size();++j){
        dfs(edge[i][j],lis,dp,maxn);
    }
}

int main(void)
{
    std::ios::sync_with_stdio(false);
    cin.tie(NULL);
    cin>>n;
    for(int i=1;i<=n;++i) cin>>a[i];
    for(int i=1;i<n;++i){
        int u,v; cin>>u>>v;
        edge[u].push_back(v);
        edge[v].push_back(u);
    }
    dfs(1,lis,dp,1);
    for(int i=1;i<=n;++i) cout<<ans[i]<<'\n';
    return 0;
}