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)

n = int(input())
a = list(map(int,input().split()))
t = [[] for _ in range(n+1)]
for _ in range(n-1):
    u,v = map(int,input().split())
    t[u].append(v)
    t[v].append(u)
    
stack = [1]
dp = [[] for _ in range(n+1)]
dp[1] = [a[0]]
ans = [1]*(n+1)
checked = []
while stack:
    b = stack.pop()
    if b in checked:
        continue
    checked.append(b)
    x = len(dp[b])
    for i in t[b]:
        if i in checked:
            continue
        stack.append(i)
        c = a[i-1]
        if dp[b][-1] < c:
            dp[i] = dp[b]+[c]
            ans[i] = x+1
        else:
            dp[i] = dp[b].copy()
            ans[i] = x
        for j in range(x-1,-1,-1):
            if j == 0 and dp[i][j] > c:
                dp[i][j] = c
            elif dp[i][j] > c and dp[i][j-1] < c:
                dp[i][j] = c
                
for i in ans[1:]:
    print(i)