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.io.*;
import java.util.*;

public class Main {

    public static class FastScanner {
        BufferedReader br;
        StringTokenizer st;

        public FastScanner(String s) {
            try {
                br = new BufferedReader(new FileReader(s));
            } catch (FileNotFoundException e) {
                // TODO Auto-generated catch block
                e.printStackTrace();
            }
        }

        public FastScanner() {
            br = new BufferedReader(new InputStreamReader(System.in));
        }

        String nextToken() {
            while (st == null || !st.hasMoreElements()) {
                try {
                    st = new StringTokenizer(br.readLine());
                } catch (IOException e) {
                    // TODO Auto-generated catch block
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }

        int nextInt() {
            return Integer.parseInt(nextToken());
        }

        long nextLong() {
            return Long.parseLong(nextToken());
        }

        double nextDouble() {
            return Double.parseDouble(nextToken());
        }
    }

    private static class Node {
        int index;

        private int value;
        List<Node> children;


        public Node(int value) {
            this.value = value;
            children = new LinkedList<>();
        }

        @Override
        public int hashCode() {
            return Objects.hash(value, index);
        }
    }

    public static void main(String[] args) {
        FastScanner fs = new FastScanner();

        int N = fs.nextInt();
        int[] a = new int[N];

        Node[] tree = new Node[N];

        Map<Node, Integer> results = new HashMap<>();
        int[] dp = new int[N];
        for (int i = 0; i < N; ++i) {
            a[i] = fs.nextInt();
            tree[i] = new Node(a[i]);
            tree[i].index = i;
            results.put(tree[i], -1);
            dp[i] = Integer.MAX_VALUE;
        }

        for (int i = 0; i < N - 1; ++i) {
            int node1 = fs.nextInt();
            int node2 = fs.nextInt();
            tree[node1 - 1].children.add(tree[node2 - 1]);
            tree[node2 - 1].children.add(tree[node1 - 1]);
        }

        Set<Node> visited = new HashSet<>();

        preorder(visited, tree[0], results, dp, 0);

        StringBuffer sb = new StringBuffer();
        for (int i = 0; i < N; ++i) {
            sb.append(results.get(tree[i]));
            sb.append(' ');
        }
        System.out.println(sb);

    }

    private static int binarySearch(int[] dp, int lookingFor) {
        int start = 0;
        int end = dp.length;

        int result = -1;
        while (start < end) {
            int middle = start + (end - start) / 2;
            if (dp[middle] >= lookingFor) {
                result = middle;
                end = middle;
            }
            else {
                start = middle + 1;
            }
        }

        return result;
    }

    private static void preorder(Set<Node> visited, Node root, Map<Node, Integer> results, int[] dp, int depth) {
        if (visited.contains(root)) {
            return;
        }

        int index = binarySearch(dp, root.value);
        int oldValue = -1;
        if (index != -1) {
            oldValue = dp[index];
            dp[index] = root.value;
        }

//        System.out.println(root.value + " " + Arrays.toString(dp));
        int foo = binarySearch(dp, Integer.MAX_VALUE - 1);
        results.put(root, Math.max(results.get(root), foo == -1? dp.length: foo));
        visited.add(root);
        for (Node child: root.children) {
            preorder(visited, child, results, dp, depth + 1);
        }

        if (index != -1) {
            dp[index] = oldValue;
        }
    }
}