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: Runtime Error)

from bisect import bisect

N = int(input())
A = list(map(int, input().split()))
X = [list(map(int, input().split())) for _ in range(N - 1)]

graph = [[] for _ in range(N)]
for u, v in X:
    graph[u - 1].append(v - 1)
    graph[v - 1].append(u - 1)

INF = 10 ** 9 + 7
dp = [INF] * N
dp[0] = -INF
ans = [0] * N

def dfs(u, p=-1):
    # Update dp
    i = bisect(dp, A[u])
    old = dp[i]
    dp[i] = A[u]

    # Calc ans
    ans[u] = i
    if p != -1:
        ans[u] = max(ans[u], ans[p])

    # Go to next
    for v in graph[u]:
        if v == p:
            continue
        dfs(v, u)

    # Restore
    dp[i] = old

dfs(0)
print(*ans)