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: Accepted)

import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;

public class Main {
	static int n;
	static List<List<Integer>> list;
	static int[] a, ans, val;

	public static void main(String[] args) throws Exception {
		BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
		n = Integer.parseInt(br.readLine());
		String[] sa = br.readLine().split(" ");
		a = new int[n];
		for (int i = 0; i < n; i++) {
			a[i] = Integer.parseInt(sa[i]);
		}
		list = new ArrayList<>(n);
		for (int i = 0; i < n; i++) {
			list.add(new ArrayList<>());
		}
		for (int i = 0; i < n - 1; i++) {
			sa = br.readLine().split(" ");
			int u = Integer.parseInt(sa[0]) - 1;
			int v = Integer.parseInt(sa[1]) - 1;
			list.get(u).add(v);
			list.get(v).add(u);
		}
		br.close();

		ans = new int[n];
		val = new int[n + 1];
		for (int i = 1; i <= n; i++) {
			val[i] = Integer.MAX_VALUE;
		}
		dfs(0, -1, 1);

		PrintWriter pw = new PrintWriter(System.out);
		for (int i = 0; i < n; i++) {
			pw.println(ans[i]);
		}
		pw.flush();
	}

	static void dfs(int x, int p, int dep) {
		int idx = Arrays.binarySearch(val, a[x]);
		if (idx < 0) idx = ~idx;
		int bk = val[idx];
		val[idx] = a[x];

		int lis = Arrays.binarySearch(val, Integer.MAX_VALUE - 1);
		if (lis < 0) {
			lis = ~lis;
			lis--;
		}
		ans[x] = lis;

		for (int next : list.get(x)) {
			if (next != p) {
				dfs(next, x, dep + 1);
			}
		}
		val[idx] = bk;
	}
}