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.util.Arrays;
import java.util.List;
import java.util.StringTokenizer;
import java.io.IOException;
import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.util.ArrayList;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 *
 * @author masterbios
 */
public class Main {
    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        InputReader in = new InputReader(inputStream);
        PrintWriter out = new PrintWriter(outputStream);
        FLISOnTree solver = new FLISOnTree();
        solver.solve(1, in, out);
        out.close();
    }

    static class FLISOnTree {
        final int MAXN = (int) (2e5) + 10;
        final int INF = (int) (1e9) + 100;
        List<Integer>[] graph;
        int[] dis;
        int[] dp;
        int[] ans;
        int n;

        public void solve(int testNumber, InputReader in, PrintWriter out) {
            n = in.nextInt();
            graph = new ArrayList[n + 1];
            dis = new int[n + 1];
            ans = new int[n + 1];
            dp = new int[MAXN];
            Arrays.fill(dp, INF);
            for (int i = 1; i <= n; i++) dis[i] = in.nextInt();
            for (int i = 0; i <= n; i++) graph[i] = new ArrayList<>();
            for (int i = 1; i < n; i++) {
                int x = in.nextInt();
                int y = in.nextInt();
                graph[x].add(y);
                graph[y].add(x);
            }
            dfs(1, 0, 1);
            for (int i = 1; i <= n; i++) out.println(ans[i]);
        }

        public void dfs(int node, int par, int depth) {
            int idx = lowerbound(1, n, dis[node]);
            int before = dp[idx];
            dp[idx] = dis[node];
//        System.out.println("depth=" + depth + " idx=" + idx + " node=" + node + " cost=" + dis[node]);
//        for (int i = 1; i <= n; i++) System.out.print(dp[i] + " ");
//        System.out.println();
            for (int child : graph[node]) {
                if(child == par) continue;
                dfs(child, node, depth + 1);
            }
            ans[node] = lowerbound(1, n, INF) - 1;
            dp[idx] = before;
        }

        public int lowerbound(int low, int high, int find) {
            int ans = -1;
            while (low <= high) {
                int mid = low + (high - low) / 2;
                if (dp[mid] == find) {
                    ans = mid;
                    high = mid - 1;
                }
                else if (dp[mid] < find) low = mid + 1;
                else high = mid - 1;
            }
            if(ans == -1) {
                ans = low;
            }
            return ans;
        }

    }

    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.hasMoreElements()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

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

    }
}