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 int n;
    static int[] a;
    static int[][] g;
    static int[] b;
    static int[] lis;
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        n = sc.nextInt();
        a = new int[n];
        for (int i = 0; i < n; i++) {
            a[i] = sc.nextInt();
        }
        g = buildGraph(sc, n, n-1);
        b = new int[n];
        Arrays.fill(b, -1);
        b[0] = 1;
        lis = new int[n+1];
        Arrays.fill(lis, Integer.MAX_VALUE);
        lis[0] = 0;
        dfs(0, 0);
        StringBuilder sb = new StringBuilder();
        for (int i = 0; i < n; i++) {
            sb.append(b[i] + "\n");
        }
        System.out.print(sb.toString());
        sc.close();

    }

    private static void dfs(int i, int max) {
        int x = a[i];
        int l = 0, r = n;
        while(1 < r - l){
            int m = (l + r) / 2;
            if(lis[m] < x) l = m;
            else r = m;
        }
        b[i] = Math.max(r, max);
        int tmp = lis[r];
        lis[r] = x;
        for(int nv : g[i]){
            if(b[nv] < 0) dfs(nv, b[i]);
        }
        lis[r] = tmp;
    }

    public static int[][] buildGraph(Scanner sc, int n, int m) {
        int[][] graph = new int[n][];
        int[][] edges = new int[m][2];
        int[] deg = new int[n];
        for (int i = 0; i < m; i++) {
            int a = sc.nextInt() - 1;
            int b = sc.nextInt() - 1;
            edges[i][0] = a;
            edges[i][1] = b;
            deg[a]++;
            deg[b]++;
        }
        for (int i = 0; i < n; i++) {
            graph[i] = new int[deg[i]];
        }
        for (int i = 0; i < m; i++) {
            int a = edges[i][0];
            int b = edges[i][1];
            graph[a][--deg[a]] = b;
            graph[b][--deg[b]] = a;
        }

        return graph;
    }

}