Problem:
Score : 600 points

Problem Statement

We have a tree with N vertices. The vertices are numbered 1 to N, and the i-th edge connects Vertex a_i and Vertex b_i.

Takahashi loves the number 3. He is seeking a permutation p_1, p_2, \ldots , p_N of integers from 1 to N satisfying the following condition:

For every pair of vertices (i, j), if the distance between Vertex i and Vertex j is 3, the sum or product of p_i and p_j is a multiple of 3.

Here the distance between Vertex i and Vertex j is the number of edges contained in the shortest path from Vertex i to Vertex j.

Help Takahashi by finding a permutation that satisfies the condition.

Constraints

2\leq N\leq 2\times 10^5

1\leq a_i, b_i \leq N

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
a_1 b_1
a_2 b_2
\vdots
a_{N-1} b_{N-1}

Output

If no permutation satisfies the condition, print -1.

Otherwise, print a permutation satisfying the condition, with space in between.
If there are multiple solutions, you can print any of them.

Sample Input 1

5
1 2
1 3
3 4
3 5

Sample Output 1

1 2 5 4 3

The distance between two vertices is 3 for the two pairs (2, 4) and (2, 5).

p_2 + p_4 = 6

p_2\times p_5 = 6

Thus, this permutation satisfies the condition.

Solution in Scala (2.11.7) (judge verdict: Runtime Error)

object Main extends App {
  val in = scala.io.Source.stdin.getLines.toSeq
  val ls = in.map(_.split(' ').map(_.toInt))
  val N = ls(0)(0)
  val input = ls.tail
  
  val nodes = (0 to N).map(e => Seq[Int]()).toArray
  input.foreach{ l =>
    val l0 = l(0)
    val l1 = l(1)
    nodes(l0) = nodes(l0) :+ l1
    nodes(l1) = nodes(l1) :+ l0
  }
  
  private def depth(i:Int, visited:Seq[Int]):Int = {
    val maxdepth = (nodes(i).map{ e =>
      val sub = if(visited.contains(e)) 0 else 1 + depth(e, (visited :+ i)) 
      if(sub>=3) return 3
      sub
    } :+ 0).max
    return maxdepth
  }
  //println(nodes.toSeq)
  //println(depth(1, Seq()))
  val all = (0 until N).map(e => depth(e, Seq()) ).max
  //println(all)
  
  if(all<3) println(-1)
  else println((1 to N).map(e => 3).mkString(" "))
}