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: Wrong Answer)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Text;
using System.Linq;

using E = System.Linq.Enumerable;

internal partial class Solver {

    List<int>[] Get3Adj(int n, List<int>[] adj) {
        var visited = new bool[n];
        var adj3 = Enumerable.Range(0, n).Select(_ => new List<int>()).ToArray();
        for (int i = 0; i < n; i++) {
            if (visited[i]) continue;
            var info = BFSUtil.BFS(adj, i);
            for (int j = 0; j < n; j++) {
                if (i == j) continue;
                if (info.Distance[j] % 3 == 0) {
                    adj3[i].Add(j);
                    adj3[j].Add(i);
                    visited[j] = true;
                }
            }
        }
        return adj3;
    }

    public void Run() {
        var n = ni();
        var a = new int[n - 1];
        var b = new int[n - 1];
        var adj = Enumerable.Range(0, n).Select(_ => new List<int>()).ToArray();
        for (int i = 0; i < n - 1; i++) {
            a[i] = ni() - 1;
            b[i] = ni() - 1;
            adj[a[i]].Add(b[i]);
            adj[b[i]].Add(a[i]);
        }
        var adj3 = Get3Adj(n, adj);
        var degree = adj3.Select(aj => aj.Count).ToArray();
        int n1 = 0, n2 = 0, n3 = 0;
        for (int i = 1; i <= n; i++) {
            var d = i % 3;
            if (d == 1) n1++;
            if (d == 2) n2++;
            if (d == 0) n3++;
        }

        var tree = new SegmentTree<Data>((d1, d2) => {
            return d1.Value < d2.Value ? d2 : d1;
        }, new Data { Value = -10000, Index = -1 });

        tree.Build(n, i => new Data { Value = degree[i], Index = i });

        var label = new int?[n];
        while (n3 > 0) {
            var maxV = tree.Query(0, n);
            var v = maxV.Index;
            foreach (var p in adj3[v]) {
                tree[p] = tree[p].Decrease();
            }
            tree[v] = new Data { Index = v, Value = -100 };
            //new { maxV }.Dump();
            label[v] = 3; n3--;
        }

        for (int i = 0; i < n; i++) {
            if (label[i].HasValue) continue;

            var queue = new Queue<int>();
            queue.Enqueue(i);
            queue.Enqueue(n1 > n2 ? 1 : 2);
            while (queue.Count > 0) {
                var pos = queue.Dequeue();
                var val = queue.Dequeue();
                if (label[pos].HasValue) continue;
                if (val == 1) n1--;
                else n2--;
                label[pos] = val;
                foreach (var p in adj3[pos]) {
                    if (label[p].HasValue) continue;
                    queue.Enqueue(p);
                    queue.Enqueue(val == 1 ? 2 : 1);
                }
            }
        }

        //new { n1, n2, n3 }.Dump();

        for (int i = 1; i <= 3; i++) {
            int cnt = 0;
            for (int j = 0; j < n; j++) {
                if (label[j] == i) {
                    label[j] += 3 * cnt;
                    cnt++;
                }
            }
        }

        cout.WriteLine(string.Join(" ", label));
    }

}

struct Data {
    public int Value, Index;
    public Data Decrease() {
        return new Data { Value = Value - 1, Index = Index };
    }
}


public sealed class SegmentTree<TMonoid> {
    public int Count { get; private set; }
    private TMonoid[] _node;
    private readonly TMonoid _identity;
    private readonly Func<TMonoid, TMonoid, TMonoid> _merge;

    public SegmentTree(Func<TMonoid, TMonoid, TMonoid> merge, TMonoid identity) {
        _merge = merge;
        _identity = identity;
    }

    public void Build(IList<TMonoid> array) {
        Build(array.Count, i => array[i]);
    }

    public void Build(int n, Func<int, TMonoid> create) {
        Count = n;

        _node = new TMonoid[2 * Count];

        for (int i = 0; i < n; i++) {
            _node[i + Count] = create(i);
        }

        for (int i = Count - 1; i > 0; i--) {
            _node[i] = _merge(_node[i << 1], _node[i << 1 | 1]);
        }
    }

    public void Set(int index, TMonoid val) {
        int p = index + Count;
        _node[p] = val;
        while (p > 0) {
            p >>= 1;
            _node[p] = _merge(_node[p << 1], _node[p << 1 | 1]);
        }
    }

    public TMonoid Query(int begin, int end) {
        TMonoid lv = _identity, rv = _identity;
        int l = begin + Count;
        int r = end + Count - 1;
        for (; l <= r; l >>= 1, r >>= 1) {
            if ((l & 1) == 1) { lv = _merge(lv, _node[l++]); }
            if ((r & 1) == 0) { rv = _merge(_node[r--], rv); }
        }
        return _merge(lv, rv);
    }

    public TMonoid this[int index] {
        get { return _node[index + Count]; }
        set { Set(index, value); }
    }
};
public class BFSTreeInfo {
    public int[] Size;
    public int[] Parent;
    public int[] Distance;
}

public static class BFSUtil {
    public static BFSTreeInfo BFS(List<int>[] adj, int root) {
        var n = adj.Length;
        var size = new int[n];
        var parents = new int[n];
        var distance = new int[n];
        var visited = new bool[n];
        var queue = new Queue<int>();
        queue.Enqueue(root);
        queue.Enqueue(-1);
        var perm = new List<int>();
        while (queue.Count > 0) {
            var current = queue.Dequeue();
            var parent = queue.Dequeue();
            if (visited[current]) continue;
            visited[current] = true;
            parents[current] = parent;
            if (parent >= 0) distance[current] = distance[parent] + 1;
            perm.Add(current);
            foreach (var next in adj[current]) {
                if (next == parent) continue;
                if (visited[next]) continue;
                queue.Enqueue(next);
                queue.Enqueue(current);
            }
        }
        perm.Reverse();
        foreach (var p in perm) {
            size[p] = 1;
            foreach (var next in adj[p]) {
                if (next == parents[p]) continue;
                size[p] += size[next];
            }
        }

        return new BFSTreeInfo {
            Size = size,
            Parent = parents,
            Distance = distance,
        };
    }

    public static int GetParent(this BFSTreeInfo[] t, int node, int root) {
        return t[root].Parent[node];
    }
    public static int GetSize(this BFSTreeInfo[] t, int node, int root) {
        return t[root].Size[node];
    }

}

// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main(string[] args) {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        using (var reader = new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length)) {
            Console.SetIn(reader);
            new Solver(Console.In, Console.Out).Run();
        }
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
    }
    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

internal static class LinqPadExtension {
    [Conditional("DEBUG")]
    public static void Dump<T>(this T obj) {
#if DEBUG
        LINQPad.Extensions.Dump(obj);
#endif
    }
}

public class Scanner {
    private readonly TextReader Reader;
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    private readonly char[] buffer = new char[2 * 1024];
    private int cursor = 0, length = 0;
    private string Token;
    private readonly StringBuilder sb = new StringBuilder(1024);

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return checked((int)NextLong()); }
    public long NextLong() {
        var s = Next();
        long r = 0;
        int i = 0;
        bool negative = false;
        if (s[i] == '-') {
            negative = true;
            i++;
        }
        for (; i < s.Length; i++) {
            r = r * 10 + (s[i] - '0');
#if DEBUG
            if (!char.IsDigit(s[i])) throw new FormatException();
#endif
        }
        return negative ? -r : r;
    }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (Token == null) {
            if (!StockToken()) {
                throw new InvalidOperationException();
            }
        }
        var token = Token;
        Token = null;
        return token;
    }

    public bool HasNext() {
        if (Token != null) {
            return true;
        }

        return StockToken();
    }

    private bool StockToken() {
        while (true) {
            sb.Clear();
            while (true) {
                if (cursor >= length) {
                    cursor = 0;
                    if ((length = Reader.Read(buffer, 0, buffer.Length)) <= 0) {
                        break;
                    }
                }
                var c = buffer[cursor++];
                if (33 <= c && c <= 126) {
                    sb.Append(c);
                } else {
                    if (sb.Length > 0) break;
                }
            }

            if (sb.Length > 0) {
                Token = sb.ToString();
                return true;
            }

            return false;
        }
    }
}