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: Accepted)

# F - LIS on Tree
import bisect
import sys
sys.setrecursionlimit(10**7)

INF = 10**18

n = int(input())
a = list(int(x) for x in input().split())
g = [[] for _ in range(n)]
for _ in range(n-1):
    u, v = map(int, input().split())
    u -= 1
    v -= 1
    g[u].append(v)
    g[v].append(u)

def dfs(now, par):
    idx = bisect.bisect_left(dp, a[now])    
    tmp = dp[idx]
    dp[idx] = a[now]
    # 親のLIS以上であれば、その値で更新
    if ans[par] <= idx:
        ans[now] = idx
    # そうでなければ親のLISを引き継ぐ
    else:
        ans[now] = ans[par]

    for i in g[now]:
        if i!=par:
            # 次にみる頂点と、自分（親）の情報を渡す
            dfs(i, now)
    dp[idx] = tmp

ans = [0] * n
dp = [-INF] + [INF] * n
dfs(0, -1)
for a in ans:
    print(a)