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 Ruby (2.7.1) (judge verdict: Accepted)

n = gets.to_i
$a = [0] + gets.split.map(&:to_i)

$road = Array.new(n+1) {[]}
(n-1).times do
    u, v = gets.split.map(&:to_i)
    $road[u] << v
    $road[v] << u
end

$dp = Array.new(n+1, 10**10)
$r = 0
$ans = Array.new(n+1)

def dfs(i, pre)
    j = $dp.bsearch_index {|e| $a[i] <= e}
    x = $dp[j]
    $dp[j] = $a[i]
    $r += 1 if j == $r
    $ans[i] = $r

    $road[i].each do |k|
        next if k == pre
        dfs(k, i)
    end

    $dp[j] = x
    $r -= 1 if x == 10**10
end

dfs(1, 0)
$ans.shift
puts $ans