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 = gets.split.map(&:to_i)
E = Hash.new { |h, k| h[k] = [] }

(N - 1).times do
  u, v = gets.split.map(&:to_i)

  E[u] << v
  E[v] << u
end

ANS = Array.new(N + 1, -Float::INFINITY)

def dfs(v, nums, visited)
  n = nums.size
  a = A[v - 1]
  visited[v] = true

  before_v = -1
  idx = nums.bsearch_index { |x| x >= a } || n

  before_v = nums[idx]
  nums[idx] = a

  ANS[v] = nums.size

  E[v].each do |u|
    next if visited[u]

    dfs(u, nums, visited)
  end

  if before_v.nil?
    nums.pop
  else
    nums[idx] = before_v
  end
end

dfs(1, [], {})

puts (1..N).map { |u| ANS[u] }