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.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.io.IOException;
import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.io.InputStream;

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, Integer[] arr) {
        if (visited[x]) return;
        visited[x] = true;
        int index = ~Arrays.binarySearch(arr, a[x], (o1, o2) -> o1>=o2 ? 1 : -1);
        int before_a_x = arr[index];
        arr[index] = a[x];
        dp[x] = ~Arrays.binarySearch(arr, 999_999_999, (o1, o2)-> o1>=o2 ? 1 : -1);

        for (Integer i : edge.get(x)) {
            if (!visited[i]) {
                children.get(x).add(i);
                parent[i] = x;
                dfs(i, arr);
            }
        }

        arr[index] = before_a_x;
    }

	public static void main(String[] args) {
		InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        InputReader in = new InputReader(inputStream);
        PrintWriter out = new PrintWriter(outputStream);
		N = in.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] = in.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 = in.nextInt()-1;
			int v = in.nextInt()-1;
            edge.get(u).add(v);
            edge.get(v).add(u);
		}

        visited = new boolean[N];
        int INF = 1_000_000_000;
        Integer[] arr = new Integer[N];
        Arrays.fill(arr, INF);
        dfs(0, arr);
        for (int i=0;i<N;i++) {
        	out.println(dp[i]);
        }

		out.close();
	}

	static class InputReader {
        public BufferedReader reader;
        public StringTokenizer tokenizer;

        public InputReader(InputStream stream) {
            reader = new BufferedReader(new InputStreamReader(stream), 32768);
            tokenizer = null;
        }

        public String next() {
            while (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

        public int nextInt() {
            return Integer.parseInt(next());
        }

        public long nextLong() {
            return Long.parseLong(next());
        }

    }
}