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 Python (3.8.2) (judge verdict: Time Limit Exceeded)

import sys
import bisect
sys.setrecursionlimit(10**8)
input = sys.stdin.readline


def inp():
    return int(input())


def inpl():
    return list(map(int, input().split()))


def main():
    def dfs(s, b, lis, ansl):

        a = aa[s]
        idx = bisect.bisect_left(lis, a)

        if len(lis) == idx:
            lis.append(a)
        else:
            lis[idx] = a

        ansl[s] = len(lis)

        if len(lines[s]) >= 3:
            for t in lines[s]:
                if t != b:
                    dfs(t, s, lis.copy(), ansl)
        else:
            for t in lines[s]:
                if t != b:
                    dfs(t, s, lis, ansl)
    N = inp()
    aa = inpl()

    lines = [[] for _ in range(N)]
    for _ in range(N - 1):
        s, t = inpl()
        s -= 1
        t -= 1
        lines[s].append(t)
        lines[t].append(s)

    lines[0].append(-1)

    ansl = [0] * N

    dfs(0, -1, [], ansl)

    print("\n".join(map(str, ansl)))


if __name__ == "__main__":
    main()