101655: [AtCoder]ABC165 F - LIS on Tree

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 $4$.