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.PrintStream;
import java.io.PrintWriter;
import java.util.*;

public class Main {
    ArrayList<Integer>[] tree;
    int[] arr;
    int[] ansArr;
    int n;
    int[] dp;
    ArrayDeque<Integer[]> q = new ArrayDeque<>();

    void run() {
        Scanner sc = new Scanner(System.in);
        n = sc.nextInt();
        arr = new int[n];
        ansArr = new int[n];
        Arrays.fill(ansArr, -1);
        dp = new int[n + 1];

        Arrays.fill(dp, Integer.MAX_VALUE);
        for (int i = 0; i < n; i++) {
            arr[i] = sc.nextInt();
        }
        tree = new ArrayList[n];
        for (int i = 0; i < n; i++) {
            tree[i] = new ArrayList<>();
        }
        for (int i = 0; i < n - 1; i++) {
            int u = sc.nextInt() - 1;
            int v = sc.nextInt() - 1;
            tree[u].add(v);
            tree[v].add(u);
        }

        dp[0] = arr[0];
        dfs(0);

        PrintWriter _out = new PrintWriter(System.out);
        for (int i = 0; i < n; i++) {
            _out.println(ansArr[i]);
        }
        _out.flush();

    }

    void dfs(int i) {
        ArrayList<Integer> nextIdxList = tree[i];
        int ans = lowerBound(dp, Integer.MAX_VALUE);
        ansArr[i] = ans;

        if (nextIdxList.isEmpty()) {
            return;
        }
        for (int nextIdx : tree[i]) {
            if (ansArr[nextIdx] != -1) {
                continue;
            }
            int next = arr[nextIdx];
            int j = lowerBound(dp, next);

            q.push(new Integer[]{j, dp[j]});
            dp[j] = next;

            dfs(nextIdx);
            Integer[] previous = q.remove();
            dp[previous[0]] = previous[1];
        }
    }


    public int upperBound(int[] array, int value) {
        int left = -1;
        int right = array.length;
        while (right - left > 1) {
            int middle = (right + left) / 2;
            if (array[middle] > value) {
                right = middle;
            } else {
                left = middle;
            }
        }
        return right;
    }

    public int lowerBound(int[] array, int value) {
        int left = -1;
        int right = array.length;
        while (right - left > 1) {
            int middle = (right + left) / 2;
            if (array[middle] >= value) {
                right = middle;
            } else {
                left = middle;
            }
        }
        return right;
    }


    void debug(Object... os) {
        System.err.println(Arrays.deepToString(os));
    }

    public static void main(String[] args) {
        new Main().run();
    }
}