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 PyPy3 (7.3.0) (judge verdict: Time Limit Exceeded)

import sys
import bisect
import collections

sys.setrecursionlimit(10 ** 8)

input = sys.stdin.readline


def main():
    N = int(input())
    A = [int(x) for x in input().split()]
    UV = [[int(x) for x in input().split()] for _ in range(N - 1)]

    T = [[] for j in range(N + 1)]

    for u, v in UV:
        T[u].append(v)
        T[v].append(u)

    s = collections.deque()

    lis = []
    s.append([1, lis])

    v = set()

    ans = [0] * (N + 1)

    while s:
        c = s[-1]
        ci, clis = c[0], c[1]

        # lis処理
        if ci not in v:
            if len(clis) == 0:
                clis.append(A[ci - 1])
            else:
                if clis[-1] < A[ci - 1]:
                    clis.append(A[ci - 1])
                else:
                    i = bisect.bisect_left(clis, A[ci - 1])
                    clis[i] = A[ci - 1]

            v.add(ci)
            ans[ci] = len(clis)

        if len(T[ci]) >= 1:
            n = T[ci].pop()
            if n in v:
                continue
            s.append([n, clis.copy()])
            continue

        s.pop()

    for i in range(1, N + 1):
        print(ans[i])


if __name__ == '__main__':
    main()