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: Time Limit Exceeded)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
using static System.Math;
using MethodImplAttribute = System.Runtime.CompilerServices.MethodImplAttribute;
using MethodImplOptions = System.Runtime.CompilerServices.MethodImplOptions;

public static class P
{
    public static void Main()
    {
        var n = int.Parse(Console.ReadLine());

        List<int>[] graph = Enumerable.Repeat(0, n).Select(_ => new List<int>()).ToArray();
        for (int i = 0; i < n - 1; i++)
        {
            var st = Console.ReadLine().Split().Select(x => int.Parse(x) - 1).ToArray();
            graph[st[0]].Add(st[1]);
            graph[st[1]].Add(st[0]);
        }

        int[] depth = Enumerable.Repeat(-1, n).ToArray();
        Stack<int> stack = new Stack<int>();
        stack.Push(0);
        depth[0] = 0;
        while (stack.Count > 0)
        {
            var elem = stack.Pop();
            foreach (var item in graph[elem])
            {
                if (depth[item] != -1) continue;
                depth[item] = depth[elem] ^ 1;
                stack.Push(item);
            }
        }
        var zeroCount = depth.Count(x => x == 0);
        var oneCount = depth.Count(x => x == 1);

        var permZeroCount = n / 3;

        int[] res = new int[n];
        
        if (zeroCount <= permZeroCount || oneCount <= permZeroCount)
        {
            var zeroClearSig = oneCount <= permZeroCount ? 1 : 0;
            HashSet<int> unused = new HashSet<int>(Enumerable.Range(1, n));
            int next = 3;
            for (int i = 0; i < depth.Length; i++)
            {
                if (depth[i] == zeroClearSig)
                {
                    res[i] = (next += 3) - 3;
                    unused.Remove(res[i]);
                }
            }
            for (int i = 0; i < depth.Length; i++)
            {
                if (res[i] == 0)
                {
                    res[i] = unused.First();
                    unused.Remove(res[i]);
                }
            }
        }
        else
        {
            int lastOne = 1;
            int lastTwo = 2;
            int lastZero = 3;
            for (int i = 0; i < depth.Length; i++)
            {
                if (depth[i] == 0) res[i] = lastOne <= n ? (lastOne += 3) - 3 : (lastZero += 3) - 3;
                else res[i] = lastTwo <= n ? (lastTwo += 3) - 3 : (lastZero += 3) - 3;
            }
        }

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