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;
 
int n,i,s,m,a,b,dist[200005],written[200005];
vector <int> v[200005],seq[200005];
queue <int> q;
 
int CeilIndex(std::vector<int>& v, int l, int r, int key) 
{ 
    while (r - l > 1) { 
        int m = l + (r - l) / 2; 
        if (v[m] >= key) 
            r = m; 
        else
            l = m; 
    } 
  
    return r; 
} 
  
int lis(std::vector<int>& v) 
{ 
    if (v.size() == 0) 
        return 0; 
  
    std::vector<int> tail(v.size(), 0); 
    int length = 1; 
  
    tail[0] = v[0]; 
    for (size_t i = 1; i < v.size(); i++) { 
  
        if (v[i] < tail[0]) 
            tail[0] = v[i]; 
        else if (v[i] > tail[length - 1]) 
            tail[length++] = v[i]; 
  
        else
            tail[CeilIndex(tail, -1, length - 1, v[i])] = v[i]; 
    } 
  
    return length; 
} 
  
 
int main() 
{
    cin>>n;
    for (int i=1; i<=n; i++)
        cin>>written[i];
        
    for (int i=1; i<=n-1; i++)
    {
        cin>>a>>b;
        v[a].push_back(b);
    }
    
    q.push(1);
    dist[1]=1;
    seq[1].push_back(written[1]);
    while (!q.empty()){
        a=q.front();
        q.pop();
        for (int i=0; i<v[a].size(); i++)
            if (dist[v[a][i]]==0) {
                dist[v[a][i]]=dist[a]+1;
                seq[v[a][i]].insert(seq[v[a][i]].end(),seq[a].begin(),seq[a].end());
                seq[v[a][i]].push_back(written[v[a][i]]);
                q.push(v[a][i]);
            }
    }
    
    for (int i=1; i<=n; i++)
    {
        cout<<lis(seq[i])<<endl;
        //for (int j=0; j<seq[i].size(); j++)
        //    cout<<seq[i][j]<<" ";
        //cout<<endl;
    }
}