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.util.*;

public class Main {
    static ArrayList<ArrayList<Integer>> graph = new ArrayList<>();;
    static int[] values;
    static int[] longest;
    static int[] ans;
    static boolean[] used;
    static int n;
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        n = sc.nextInt();
        values = new int[n];
        for (int i = 0; i < n; i++) {
            values[i] = sc.nextInt();
            graph.add(new ArrayList<>());
        }
        longest = new int[n + 1];
        Arrays.fill(longest, Integer.MAX_VALUE);
        longest[0] = 0;
        ans = new int[n];
        used = new boolean[n];
        for (int i = 0; i < n - 1; i++) {
            int a = sc.nextInt() - 1;
            int b = sc.nextInt() - 1;
            graph.get(a).add(b);
            graph.get(b).add(a);
        }
        search(0, 0);
        StringBuilder sb = new StringBuilder();
        for (int x : ans) {
            sb.append(x).append("\n");
        }
        System.out.print(sb);
    }
    
    static void search(int idx, int max) {
        if (used[idx]) {
            return;
        }
        used[idx] = true;
        int left = 0;
        int right = n + 1;
        while (right - left > 1) {
            int m = (left + right) / 2;
            if (longest[m] < values[idx]) {
                left = m;
            } else {
                right = m;
            }
        }
        int save = longest[left + 1];
        max = Math.max(max, left + 1);
        longest[left + 1] = Math.min(longest[left + 1], values[idx]);
        ans[idx] = max;
        for (int x : graph.get(idx)) {
            search(x, max);
        }
        longest[left + 1] = save;
    }
}