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: Time Limit Exceeded)

require 'set'; require 'prime'; require 'pp'
INF=Float::INFINITY; MOD=10**9+7
n = gets.chomp.to_i
a = gets.chomp.split.map(&:to_i)
edges = Array.new(n-1)
ans = Array.new(n)
(n-1).times do
  x, y = gets.chomp.split.map(&:to_i)
  x -= 1
  y -= 1
  edges[x] ||= []
  edges[y] ||= []
  edges[x].push y
  edges[y].push x
end
 
dp = Array.new()
dp.push a[0]
 
q = [[0, dp]]
ans[0] = dp.size
visited = Set.new([0])
until q.empty?
  index, dp = q.pop
  edge_size = edges[index].size
  edges[index].each_with_index do |v, i|
    next if visited.include? v
    visited.add v
 
    new_dp = i == edge_size - 1 ? dp : dp.dup
    val = a[v]
    bi = dp.bsearch_index{|x| x >= val}
    if bi.nil?
      new_dp.push val
    else
      new_dp[bi] = val
    end
    ans[v] = new_dp.size
    q.push [v, new_dp]
  end
end
ans.each do |line|
  puts line
end