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 C# (Mono 4.6.2.0) (judge verdict: Runtime Error)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using static System.Console;
using static System.Math;

namespace AtTest.ForYellow
{
    class Hitachi2020_C
    {
        static void Main(string[] args)
        {
            var sw = new System.IO.StreamWriter(OpenStandardOutput()) { AutoFlush = false };
            SetOut(sw);

            Method(args);

            Out.Flush();
        }

        static void Method(string[] args)
        {
            int n = ReadInt();
            List<int>[] graph = new List<int>[n];
            for(int i = 0; i < n; i++)
            {
                graph[i] = new List<int>();
            }

            for(int i = 0; i < n - 1; i++)
            {
                int[] ab = ReadInts();
                int a = ab[0] - 1;
                int b = ab[1] - 1;
                graph[a].Add(b);
                graph[b].Add(a);
            }

            var numbers = new HashSet<int>[3];
            for(int i = 0; i < 3; i++)
            {
                numbers[i] = new HashSet<int>();
            }
            for(int i = 1; i <= n; i++)
            {
                numbers[i % 3].Add(i);
            }

            int[] res = new int[n];
            Queue<int[]> que = new Queue<int[]>();
            que.Enqueue(new int[3] { 0, 1, -1 });
            while (que.Count > 0)
            {
                int[] val = que.Dequeue();
                int idx = val[0];
                int color = val[1];
                int par = val[2];

                if (numbers[color].Count > 0)
                {
                    int num = numbers[color].First();
                    res[idx] = num;
                    numbers[color].Remove(num);
                }
                else
                {
                    int num = numbers[0].First();
                    res[idx] = num;
                    numbers[0].Remove(num);
                }

                for (int i = 0; i < graph[idx].Count; i++){
                    int to = graph[idx][i];
                    if (to == par)
                    {
                        continue;
                    }

                    que.Enqueue(new int[3] { to, color == 1 ? 2 : 1, idx });
                }
            }

            WriteLine(string.Join(" ", res));
        }

        private static string Read() { return ReadLine(); }
        private static char[] ReadChars() { return Array.ConvertAll(Read().Split(), a => a[0]); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
    }
}