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: Wrong Answer)

import java.util.*;
import java.io.*;
import java.nio.charset.StandardCharsets;

public class Main {
	int N;
	Tuple[] tree;

	static class Tuple {
		int id;
		long a;
		Set<Integer> conns = new HashSet<>();
	}

	static class Pair {
		int leaf;
		int num;
		long max;
	}

	public static void main(String[] args) throws IOException {
		InputStreamReader reader = new InputStreamReader(System.in, StandardCharsets.UTF_8);
		BufferedReader in = new BufferedReader(reader);
		Main ins = new Main(in);
		ins.calc();
		ins.showResult();
	}

	Main(BufferedReader in) throws IOException {
		String[] tokens = in.readLine().split(" ");
		N = Integer.parseInt(tokens[0]);
		tokens = in.readLine().split(" ");
		tree = new Tuple[N];
		for (int i = 0; i < N; ++i) {
			Tuple tuple = new Tuple();
			tuple.id = i;
			tuple.a = Long.parseLong(tokens[i]);
			tree[i] = tuple;
		}
		for (int i = 0; i < N - 1; ++i) {
			tokens = in.readLine().split(" ");
			int u = Integer.parseInt(tokens[0]) - 1;
			int v = Integer.parseInt(tokens[1]) - 1;
			tree[u].conns.add(v);
			tree[v].conns.add(u);
		}
	}

	Map<Integer, Pair> search(int current, int parent) {
		Set<Integer> set = new HashSet<>();
		set.addAll(tree[current].conns);
		set.remove(parent);
		if (set.isEmpty()) {
			Map<Integer, Pair> map = new HashMap<>();
			Pair pair = new Pair();
			pair.leaf = current;
			pair.max = tree[current].a;
			pair.num = 1;
			map.put(current, pair);
			return map;
		} else {
			// mapをマージする
			Map<Integer, Pair> map = new HashMap<>();
			for (Integer child : set) {
				Map<Integer, Pair> childMap = search(child, current);
				for (Integer key : childMap.keySet()) {
					map.put(key, childMap.get(key));
				}
			}
			for (Integer key : map.keySet()) {
				if (map.get(key).max > tree[current].a) {
					map.get(key).num++;
					map.get(key).max = tree[current].a;
				}
			}
			Pair pair = new Pair();
			pair.leaf = current;
			pair.max = tree[current].a;
			pair.num = 1;
			map.put(current, pair);
			return map;
		}
	}

	void calc() {
		Map<Integer, Pair> map = search(0, -1);
		List<Integer> keys = new ArrayList<>();
		keys.addAll(map.keySet());
		Collections.sort(keys);
		for (int i = 0; i < keys.size(); ++i) {
			System.out.println(map.get(keys.get(i)).num);
		}
	}

	void showResult() {
	}

}