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 Java8 (OpenJDK 1.8.0) (judge verdict: Runtime Error)

import java.util.Scanner;
public class Main {
  public static void main (String args[]) {
    Scanner sc = new Scanner(System.in);
    int N = sc.nextInt();
    int three = N/3;
    int tree[] = new int[N];
    int from, to;
    int ki, gu;
    ki = gu = 0;
    while(sc.hasNext()){
      from = sc.nextInt();
      to = sc.nextInt();
      if(tree[from-1]==2){
        if(tree[to-1]==0){
          tree[to-1] = 1;
          ki++;
        }
      } else if(tree[from-1]==1){
        if(tree[to-1]==0){
          tree[to-1] = 2;
          gu++;
        }
      } else {
        tree[from-1]=2;
        if(tree[to-1]==0){
          tree[to-1] = 1;
          ki++;
        }
      }
    }
    int ans[] = new int[N];
    boolean check[] = new boolean[N];
    int k, l, m;
    k = l = 0;
    m = 1;
    if(three<ki&&three<gu){
      for(int i=0;i<N;i++){
        if(tree[i]==1&&k<three) {
          ans[i]=3*k+1;
          check[ans[i]-1]=true;
          k++;
        }
        else if(tree[i]==2&&l<three){
          ans[i]=3*l+2;
          check[ans[i]-1]=true;
          l++;
        }
      }
    }
    else if(ki<three){
      for(int i=0;i<N;i++){
        if(tree[i]==1) {
            ans[i]=3*m;
            check[ans[i]-1]=true;
            m++;
        }
      }
    }
    else if(gu<three){
      for(int i=0;i<N;i++){
        if(tree[i]==2) {
            ans[i]=3*m;
            check[ans[i]-1]=true;
            m++;
        }
      }
    }
    int c = 0;
    for(int i=0;i<N;i++){
      System.err.print(tree[i]);
      if(ans[i]!=0){
        System.out.print(ans[i]+" ");
      } else {
        while(check[c]){ 
          c++;
        }
        System.out.print((c+1)+" ");
        c++;
      }
    }
  }
}