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.*;
import java.io.PrintWriter;
public class Main{
	static ArrayList<Integer>[] list;
	static int[] a;
	static int[] dp;
	static int[] ans;
	public static void main(String[] args){
		Scanner sc=new Scanner(System.in);
		int n=sc.nextInt();
		list=new ArrayList[n];
		a=new int[n];
		dp=new int[n+1];
		ans=new int[n];
		Arrays.fill(dp,1000000001);
		dp[0]=0;
		for(int i=0; i<n; i++){
			a[i]=sc.nextInt();
			list[i]=new ArrayList<>();
		}
		for(int i=1; i<n; i++){
			int u=sc.nextInt()-1;
			int v=sc.nextInt()-1;
			list[u].add(v);
			list[v].add(u);
		}
		dfs(0,0);
		PrintWriter pw=new PrintWriter(System.out);
		for(int i=0; i<n; i++){
			pw.println(ans[i]);
		}
		pw.flush();
	}
	static void dfs(int now,int from){
		int idx=Arrays.binarySearch(dp,a[now]);
		if(idx<0){
			idx=-idx-1;
		}
		int tmp=idx;
		int val=dp[idx];
		dp[idx]=a[now];
		ans[now]=idx;
		
		for(int i:list[now]){
			if(i!=from){
				dfs(i,now);
			}
		}
		dp[tmp]=val;
	}
}