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: Wrong Answer)

lines = []
while line = gets
    lines << line.chomp.split(' ').map(&:to_i)
end

N = lines[0][0]

a = []
for i in 1..N do
  a.push(lines[1][i-1])
end

accessible = []
for i in 1..N do
  accessible[i-1] = []
end
for i in 1..N-1 do   #loop as regard to paths
  from = lines[i+1][0]
  to = lines[i+1][1]
  accessible[from-1].push(to)
  accessible[to-1].push(from)
end

finished = []
num_array = []
max_num_array = []
for i in 1..N do
  finished[i-1] = false
  num_array[i-1] = 1
  max_num_array[i-1] = 1
end

targets_list = [1]
finished[0] = true
while targets_list != [] do
  next_targets = []
  targets_list.each do |i|
    accessible[i-1].each do |j|
      if finished[j-1] == false then
        if a[j-1] > a[i-1] then
          num_array[j-1] = num_array[i-1] + 1
          max_num_array[j-1] = max_num_array[i-1] + 1
        else
          num_array[j-1] = 0
          max_num_array[j-1] = max_num_array[i-1]
        end
        next_targets.push(j)
        finished[j-1] = true
      end
    end
  end
  targets_list = next_targets
end

puts max_num_array