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: Runtime Error)

import java.util.ArrayList;
import java.util.Arrays;
import java.util.HashSet;
import java.util.List;
import java.util.Scanner;
import java.util.Set;


public class Main {
    static List<Integer>[] tree;
    static int[] changedDpIndex;
    static int[] changedValue;
    static int[] dpValue;
    static int[] ans;
    static int[] value;
    public static void main(String [] args) {
        Scanner scanner = new Scanner(System.in);
        int n = scanner.nextInt();
        value = new int[n + 1];
        Set<Integer> valueAppeared = new HashSet<>();
        for (int i = 1; i <= n; ++i) {
            value[i] = scanner.nextInt();
            valueAppeared.add(value[i]);
        }
        tree = new List[n + 1];
        for (int i = 1; i <= n; ++i) {
            tree[i] = new ArrayList<>();
        }
        for (int i = 1; i < n; ++i) {
            int u = scanner.nextInt();
            int v = scanner.nextInt();
            tree[u].add(v);
            tree[v].add(u);
        }
        ans = new int[n + 1];
        ans[1] = 1;
        dpValue = new int[valueAppeared.size()];
        Arrays.fill(dpValue, Integer.MAX_VALUE);
        changedDpIndex = new int[n + 1];
        changedValue = new int[n + 1];
        for (int i = 1; i <= n; ++i) {
            changedDpIndex[i] = -1;
            changedValue[i] = Integer.MAX_VALUE;
        }
        dfs(1, 0, 0);
        for (int k = 1; k <= n; ++k) {
            System.out.println(ans[k]);
        }
    }
    public static void dfs(
            int currentIndex,
            int father,
            int maxPos
    ) {
        int binarySearchIndex = Arrays.binarySearch(dpValue, 0, maxPos, value[currentIndex]);
        int length;
        if (binarySearchIndex >= 0 && binarySearchIndex < dpValue.length) {
            // nodes[currentIndex] = new Node(currentIndex, -1, -1);
            length = ans[father];
        } else {
            int insertPos = -1 - binarySearchIndex;
            changedDpIndex[currentIndex] = insertPos;
            changedValue[currentIndex] = dpValue[insertPos];
            if (dpValue[insertPos] == Integer.MAX_VALUE) {
                length = insertPos + 1;
            } else {
                length = ans[father];
            }
            dpValue[insertPos] = value[currentIndex];
        }
        //  length = -Arrays.binarySearch(dpValue, Integer.MAX_VALUE - 1) - 1;
        ans[currentIndex] = length;
        maxPos = Math.max(maxPos, length + 1);

        // visit sons
        for (int son: tree[currentIndex]) {
            if (son == father) {
                continue;
            }
            dfs(son, currentIndex, maxPos);
        }

        if (changedDpIndex[currentIndex] != -1) {
            // this node replaced some node before, recover it
            dpValue[changedDpIndex[currentIndex]] = changedValue[currentIndex];
            //    nodes[currentIndex] = new Node(currentIndex, -1, -1);
        }

    }
}