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: Wrong Answer)

import sys
from bisect import bisect_left as bl
readline = sys.stdin.readline

def euler_tour(root, Edge):
    N = len(Edge)
    stack = [root]
    used = [False]*N
    tour = []
    down = []
    dist = [0]*N
    ta = tour.append
    sa = stack.append
    St = [None]*N
    En = [None]*N
    cnt = -1
    while stack:
        cnt += 1
        vn = stack.pop()
        ta(vn)
        
        if St[vn] is None:
            St[vn] = cnt
        En[vn] = cnt
        if used[vn]:
            continue
        for vf in Edge[vn]:
            if used[vf]:
                sa(vf)
            else:
                dist[vf] = 1+dist[vn]
                down.append(vf)
        stack.extend(down)
        down = []
        used[vn] = True
    return St, En, tour, dist





N = int(readline())
A = list(map(int, readline().split()))
Edge = [[] for _ in range(N)]
for _ in range(N-1):
    a, b = map(int, readline().split())
    a -= 1
    b -= 1
    Edge[a].append(b)
    Edge[b].append(a)

_, _, L, dist = euler_tour(0, Edge) 

print(L)
table = [None]*N
table[0] = (-1, -1)
dp = [A[0]]
pre = 0
Ans = [None]*N
for l in L[1:]:
    al = A[l]
    if dist[pre] > dist[l]:
        x, j = table[pre]
        if x == -1:
            dp.pop()
        else:
            dp[j] = x
    else:
        if dp[-1] < al:
            dp.append(al)
            table[l] = (-1, -1)
        else:
            idx = bl(dp, al)
            x = dp[idx]
            dp[idx] = al
            table[l] = (x, idx)
    pre = l
    Ans[l] = len(dp)
        
print('\n'.join(map(str, Ans)))