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 Java (OpenJDK 11.0.6) (judge verdict: Wrong Answer)

import java.util.*;
class Main{
  public static void main(String args[]){
    Scanner sc = new Scanner(System.in);
    int n = sc.nextInt();
    int a[]=new int[n];
    int lis[]=new int[n];
    ArrayList<ArrayList<Integer>> g = new ArrayList<ArrayList<Integer>>();
    int ans[]=new int[n];
    for(int i=0;i<n;i++){
      a[i]=sc.nextInt();
      lis[i]=-1;
      g.add(new ArrayList<Integer>());
    }
    for(int i=0;i<n-1;i++){
      int v1=sc.nextInt(),v2=sc.nextInt();
      g.get(v1-1).add(v2-1);
      g.get(v2-1).add(v1-1);
    }

    dfs(g,0,-1,a,lis,ans);
    for(int i=0;i<n;i++){
      pl(ans[i]+"");
    }

  }

  public static void dfs(ArrayList<ArrayList<Integer>> g,int v,int pr,int[]a,int[]lis,int[]ans){
    int x[]=liss(a[v],lis);
    ans[v]=x[0]+1;
    for(int i:g.get(v)){
      if(i!=pr){
        dfs(g,i,v,a,lis,ans);
      }
    }
    lis[x[0]]=x[1];
  }

  public static int[] liss(int n,int[]lis){
    for(int i=0;i<lis.length;i++){
      if(lis[i]==-1||lis[i]>=n){
        int k=lis[i];
        lis[i]=n;
        return new int[]{i,k};
      }
    }
    return new int[]{-1};
  }

  public static void pr(String str){
    System.out.print(str);
  }
  public static void pl(String str){
    System.out.println(str);
  }
}