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
from bisect import bisect_left

read = sys.stdin.read
readline = sys.stdin.readline


def main():
    N = int(readline())
    a = tuple(map(int, readline().split()))
    uv = map(int, read().split())
    graph = [[] for _ in range(N)]
    for u, v in zip(uv, uv):
        u -= 1
        v -= 1
        graph[u].append(v)
        graph[v].append(u)

    answer = [-1] * N
    stack = [(0, [a[0]])]
    answer[0] = 1

    while stack:
        V, LIS = stack.pop()
        if V == 0:
            for v in graph[V]:
                if answer[v] != -1:
                    continue
                tmp=a[v]
                if tmp > LIS[-1]:
                    new_LIS = LIS[:] + [tmp]

                else:
                    new_LIS = LIS[:]
                    new_LIS[bisect_left(new_LIS, tmp)] = tmp
                answer[v] = len(new_LIS)
                stack.append((v, new_LIS))
            continue
        if len(graph[V]) > 2:
            for v in graph[V]:
                tmp=a[v]
                if answer[v] != -1:
                    continue
                if tmp > LIS[-1]:
                    new_LIS = LIS[:] + [tmp]

                else:
                    new_LIS = LIS[:]
                    new_LIS[bisect_left(new_LIS, tmp)] = tmp
                answer[v] = len(new_LIS)
                stack.append((v, new_LIS))
        else:
            for v in graph[V]:
                tmp=a[v]
                if answer[v] != -1:
                    continue
                if tmp > LIS[-1]:
                    LIS += [tmp]
                else:
                    LIS[bisect_left(LIS, tmp)] = tmp
                answer[v] = len(LIS)
                stack.append((v, LIS))

    print('\n'.join(map(str, answer)))


if __name__ == '__main__':
    main()