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# (Mono-mcs 6.8.0.105) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
class Program
{
	static void Main(string[] args)
	{
		int N = int.Parse(Console.ReadLine());
		int[]A = new int[N];
		string[] str = Console.ReadLine().Split();
		for(var i=0;i<N;i++){
			A[i] = int.Parse(str[i]);
		}
		var list = new List<int>[N];
		for(var i=0;i<N;i++){
			list[i] = new List<int>();
		}
		for(var i=0;i<N-1;i++){
			string[] str2 = Console.ReadLine().Split();
			int a1 = int.Parse(str2[0])-1;
			int a2 = int.Parse(str2[1])-1;
			list[a1].Add(a2);
			list[a2].Add(a1);
		}
		int[]C = new int[N];
		int[]D = new int[N];
		var listC = new List<int>[N];
		var listA = new List<int>();
		listA.Add(0);
		for(var i=0;i<N;i++){
			C[i] = -1;
			D[i] = -1;
			listC[i] = new List<int>();
		}
		C[0] = 1;
		D[0] = 1;
		listC[0].Add(1);
		while(listA.Count>0){
			var listB = new List<int>(listA);
			listA.Clear();
			foreach(var j in listB){
				foreach(var k in list[j]){
					if(C[k] == -1){
						int min = 0;
						int max = listC[j].Count;
						while(min<max){
							int m = (min+max)/2;
							if(listC[j][m] >= A[k]){
								max = m;
							} else {
								min = m+1;
							}
                          //Console.WriteLine(m+" "+min+" "+max);
						}
                  		listC[k] = new List<int>(listC[j]);
						if(min==listC[j].Count){
							listC[k].Add(A[k]);
						} else {
							listC[k][min] = A[k];
						}
						C[k] = 1;
						listA.Add(k);
					}
				}
			}
		}
		for(var i=0;i<N;i++){
			Console.WriteLine(listC[i].Count);
		}
	}
}