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 Julia (1.4.0) (judge verdict: Accepted)

function main()
	N=parse(Int,readline())
	A=map(x->parse(Int,x),split(readline()))
	G=[Int[] for _=1:N]
	for i=2:N
		u,v=map(x->parse(Int,x),split(readline()))
		push!(G[u],v)
		push!(G[v],u)
	end
	dp=ones(Int,N+1)*1145141919
	dp[1]=0
	ans=zeros(Int,N)
	function dfs(u,p,R)
		l=1
		r=R
		while r-l>1
			m=div(l+r,2)
			if dp[m]<A[u]
				l=m
			else
				r=m
			end
		end
		pre=dp[r]
		dp[r]=A[u]
		if R==r
			R+=1
		end
		ans[u]=R-2
		for v=G[u]
			if v!=p
				dfs(v,u,R)
			end
		end
		dp[r]=pre
	end
	dfs(1,0,2)
	println.(ans)
end
main()