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: Time Limit Exceeded)

import java.util.*;

public class Main {
	static int N;
	static int[] a;
    static ArrayList<ArrayList<Integer>> edge;
    static ArrayList<ArrayList<Integer>> children;
    static int[] parent;
    static boolean[] visited;
    static int[] dp;

    static void dfs(int x, ArrayList<Integer> list) {
        if (visited[x]) return;
        visited[x] = true;
        int index = ~Collections.binarySearch(list, a[x], (o1, o2) -> o1>=o2 ? 1 : -1);
        if (index<list.size()) list.set(index, a[x]);
        else list.add(a[x]);
        dp[x] = list.size();

        for (Integer i : edge.get(x)) {
        	if (edge.get(x).size()>1) {
        		ArrayList<Integer> list_copy = new ArrayList<Integer>(list);
	            if (!visited[i]) {
	                children.get(x).add(i);
	                parent[i] = x;
	                dfs(i, list_copy);
	            }
        	} else {
	            if (!visited[i]) {
	                children.get(x).add(i);
	                parent[i] = x;
	                dfs(i, list);
	            }
        	}
        }
    }

	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		N = sc.nextInt();
		a = new int[N];
        edge = new ArrayList<ArrayList<Integer>>();
        children = new ArrayList<ArrayList<Integer>>();
        parent = new int[N];
        parent[0] = -1;
        dp = new int[N];
		for (int i=0;i<N;i++) {
			a[i] = sc.nextInt();
		}
        for (int i=0;i<N;i++) {
            ArrayList<Integer> add_1 = new ArrayList<Integer>();
            ArrayList<Integer> add_2 = new ArrayList<Integer>();
            edge.add(add_1);
            children.add(add_2);
        }

		for (int i=0;i<N-1;i++) {
			int u = sc.nextInt()-1;
			int v = sc.nextInt()-1;
            edge.get(u).add(v);
            edge.get(v).add(u);
		}

        visited = new boolean[N];
        ArrayList<Integer> list = new ArrayList<Integer>();
        dfs(0, list);
        for (int i=0;i<N;i++) {
        	System.out.println(dp[i]);
        }
        // System.out.println(edge);
        // System.out.println(children);
	}
}