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: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;
const int maxn = 200000;
const int inf = 1000000001;
int n;
vector<int> a(maxn);
vector<vector<int>> edge(maxn);
vector<int> ans(maxn);

void dfs(int from, int now, vector<int> lis){
    auto p = upper_bound(lis.begin(),lis.end(),a[now]);
    *p = a[now];
    ans[now] = lower_bound(lis.begin(),lis.end(),inf) -lis.begin();
    for(int i=0; i<edge[now].size(); i++){
        if(edge[now][i]==from) continue;
        dfs(now,edge[now][i],lis);
    }
}

int main(){
    cin >> n;
    for(int i=0; i<n; i++) cin >> a[i];
    for(int i=0; i<n-1; i++){
        int u,v;
        cin >> u >> v;
        u--;v--;
        edge[u].push_back(v);
        edge[v].push_back(u);
    }
    ans[0] = 1;
    for(int i=0; i<edge[0].size(); i++){
        vector<int> lis(n,inf);
        lis[0] = a[0];
        dfs(0,edge[0][i],lis);
    }
    for(int i=0; i<n; i++) cout << ans[i] << endl;
    return 0;
}