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 Scala (2.13.1) (judge verdict: Time Limit Exceeded)

import scala.collection.mutable
import scala.annotation.tailrec
object Main extends App {

  import scala.io.StdIn._
  val n = readInt
  val nodeValues = readLine.trim.split(' ').map(_.toInt)
  val nodes = Array.fill(n){Nil: List[Int]}
  for (_ <- 1 until n) {
    val Array(u, v) = readLine.trim.split(' ').map(_.toInt - 1)
    nodes(u) ::= v
    nodes(v) ::= u
  }
  val history = Array.fill(n){Vector[Int]()}
  history(0) = Vector(nodeValues(0))
  val result = Array.fill(n){0}
  setHistory((List(0)))
  result.foreach(println)
  @tailrec
  def setHistory(stack: List[Int]): Unit = {
    stack match {
      case Nil => ()
      case h::rest =>
        val vec = history(h)
        var left = 0
        var right = vec.length
        while (left < right) {
          val mid = (left + right) >> 1
          if (vec(mid) < nodeValues(h)) {
            left = mid + 1
          } else {
            right = mid
          }
        }
        val nextVec = if (right < vec.length) vec.updated(right, nodeValues(h)) else vec.appended(nodeValues(h))
        result(h) = nextVec.length
        var next = rest
        for (n <- nodes(h) if history(n).isEmpty) {
          next ::= n
          history(n) = nextVec
        }
        setHistory(next)
    }
  }
}