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: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Runtime.CompilerServices;
using System.Text;
using System.Threading.Tasks;

namespace AtCorder
{
	public class Program
	{
		static void Main(string[] args)
		{
			var cin = new Scanner2();
			int n = cin.Int();
			var to = new List<int>[n];
			for (int i = 0; i < n; i++) {
				to[i] = new List<int>();
			}

			for (int i = 0; i < n - 1; i++) {
				int a = cin.Int() - 1;
				int b = cin.Int() - 1;
				to[a].Add(b);
				to[b].Add(a);
			}

			var d = new int[n];
			for (int i = 0; i < n; i++) {
				d[i] = int.MaxValue;
			}

			d[0] = 0;
			var q = new Queue<int>();
			q.Enqueue(0);
			while (q.Count > 0) {
				int cur = q.Dequeue();
				foreach (int next in to[cur]) {
					if (d[next] != int.MaxValue) {
						continue;
					}

					d[next] = d[cur] + 1;
					q.Enqueue(next);
				}
			}

			var s1 = new List<int>(n);
			var s2 = new List<int>(n);
			var q0 = new Queue<int>(n / 3);
			var q1 = new Queue<int>(n / 3);
			var q2 = new Queue<int>(n / 3);
			for (int i = 0; i < n; i++) {
				if (d[i] % 2 == 0) {
					s1.Add(i);
				} else {
					s2.Add(i);
				}

				if ((i + 1) % 3 == 0) {
					q0.Enqueue(i + 1);
				} else if ((i + 1) % 3 == 1) {
					q1.Enqueue(i + 1);
				} else {
					q2.Enqueue(i + 1);
				}
			}

			var ans = new int[n];
			int min = Math.Min(s1.Count, s2.Count);
			if (min <= n / 3) {
				List<int> sf = s1.Count < s2.Count ? s1 : s2;
				List<int> ss = s1.Count < s2.Count ? s2 : s1;
				foreach (int t in sf) {
					ans[t] = q0.Dequeue();
				}

				foreach (int t in ss) {
					if (q0.Count > 0) {
						ans[t] = q0.Dequeue();
					} else if (q1.Count > 0) {
						ans[t] = q1.Dequeue();
					} else {
						ans[t] = q2.Dequeue();
					}
				}
			} else {
				foreach (int t in s1) {
					if (q1.Count > 0) {
						ans[t] = q1.Dequeue();
					} else {
						ans[t] = q0.Dequeue();
					}
				}

				foreach (int t in s2) {
					if (q2.Count > 0) {
						ans[t] = q2.Dequeue();
					} else {
						ans[t] = q0.Dequeue();
					}
				}
			}

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

	public class Scanner2
	{
		private readonly char[] delimiter_ = new char[] { ' ' };
		private readonly string filePath_;
		private readonly Func<string> reader_;
		private string[] buf_;
		private int index_;

		public Scanner2(string file = "")
		{
			if (string.IsNullOrWhiteSpace(file)) {
				reader_ = Console.ReadLine;
			} else {
				filePath_ = file;
				var fs = new StreamReader(file);
				reader_ = fs.ReadLine;
			}
			buf_ = new string[0];
			index_ = 0;
		}

		public string Next()
		{
			if (index_ < buf_.Length) {
				return buf_[index_++];
			}

			string st = reader_();
			while (st == "") {
				st = reader_();
			}

			buf_ = st.Split(delimiter_, StringSplitOptions.RemoveEmptyEntries);
			if (buf_.Length == 0) {
				return Next();
			}

			index_ = 0;
			return buf_[index_++];
		}

		public int Int() => int.Parse(Next());
		public long Long() => long.Parse(Next());
		public double Double() => double.Parse(Next());

		public int[] ArrayInt(int N, int add = 0)
		{
			int[] Array = new int[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Int() + add;
			}
			return Array;
		}

		public long[] ArrayLong(int N, long add = 0)
		{
			long[] Array = new long[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Long() + add;
			}
			return Array;
		}

		public double[] ArrayDouble(int N, double add = 0)
		{
			double[] Array = new double[N];
			for (int i = 0; i < N; i++) {
				Array[i] = Double() + add;
			}
			return Array;
		}

		public void Save(string text)
		{
			if (string.IsNullOrWhiteSpace(filePath_)) {
				return;
			}

			File.WriteAllText(filePath_ + "_output.txt", text);
		}
	}
}