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.Linq;
using System.Collections.Generic;

namespace Contest
{
    class Program
    {
        static void Main(string[] args)
        {
            HitachiC();
        }
        static void HitachiA()
        {
            var s = ReadStr();
            if (s.Length % 2 == 1) { Console.WriteLine("No");return; }
            for(int i = 0; i < s.Length; ++i)
            {
                if (s[i] != (i%2 == 0 ? 'h' : 'i'))
                {
                    Console.WriteLine("No"); return;
                }
            }
            Console.WriteLine("Yes");
        }
        static void HitachiB()
        {
            var abm = ReadIntN();
            var A = abm[0];
            var B = abm[1];
            var M = abm[2];
            var a = ReadIntN();
            var b = ReadIntN();
            var xyc = new List<int[]>();
            for(int i = 0; i < M; ++i)
            {
                xyc.Add(ReadIntN());
            }
            var naive = a.Min() + b.Min();
            foreach(var cpn in xyc)
            {
                var v = a[cpn[0]-1] + b[cpn[1]-1] - cpn[2];
                naive = Math.Min(naive, v);
            }
            Console.WriteLine(naive);
        }
        static void HitachiC()
        {
            var N = ReadInt();
            var Adj = ReadAdjacencyList(N, N - 1);

            var dist = new int[N + 1];
            dist[1] = 0;
            List<int> list = new List<int>();
            bool[] check = new bool[N + 1];
            check[1] = true;
            list.Add(1);
            int current = 1;
            while (list.Count > 0)
            {
                var next = new List<int>();
                foreach(var v in list)
                {
                    foreach(var w in Adj[v])
                    {
                        if (!check[w])
                        {
                            check[w] = true;
                            dist[w] = current;
                            next.Add(w);
                        }
                    }
                }
                list = next;
            }
            var mod = new int[] { 0, 1, 2, 0, 2, 1 };
            var count = new int[3];
            for(int i = 1; i <= N; ++i)
            {
                count[mod[dist[i] % 6]]++;
            }
            var order = Enumerable.Range(0, 3).OrderBy(i => count[i]).ToArray();
            var place = new int[N + 1];
            int m1 = 1;
            int m2 = 2;
            int m0 = 3;
            if (count[order[2]] > N * 2 / 3)
            {
                for (int i = 1; i <= N; ++i)
                {
                    var x = mod[mod[dist[i] % 6]];
                    if (order[2] == x)
                    {
                        if (m1 > N)
                        {
                            if (m2 > N)
                            {
                                place[i] = m0; m0 += 3;
                            }
                            else
                            {
                                place[i] = m2; m2 += 3;
                            }
                        }
                        else
                        {
                            place[i] = m1; m1 += 3;
                        }
                    }
                    else
                    {
                        place[i] = m0; m0 += 3;
                    }
                }
            }
            else
            {
                for (int i = 1; i <= N; ++i)
                {
                    var x = mod[mod[dist[i] % 6]];
                    if (order[1] == x)
                    {
                        if (m1 > N)
                        {
                            place[i] = m0; m0 += 3;
                        }
                        else
                        {
                            place[i] = m1; m1 += 3;
                        }
                    }
                    else if (order[2] == x)
                    {
                        if (m2 > N)
                        {
                            place[i] = m0; m0 += 3;
                        }
                        else
                        {
                            place[i] = m2; m2 += 3;
                        }
                    }
                    else
                    {
                        place[i] = m0; m0 += 3;
                    }
                }
            }
            var ans = new int[N];
            for(int i = 1; i <= N; ++i)
            {
                ans[place[i] - 1] = i;
            }
            Console.WriteLine(String.Join(" ", ans));
        }
        static void HitachiD()
        {
        }
        static void HitachiE()
        {

        }


        class Util
        {
            public static readonly long P = 1000000007;

            public static long Sum(long[] array)
            {
                return Sum(array, array.Length);
            }
            public static long Sum(long[] array, int n)
            {
                long total = 0;
                for(int i = 0; i < n; ++i)
                {
                    total += array[i];
                }
                return total;
            }
            public static long Count(long[] array, long x)
            {
                return Count(array, x, array.Length);
            }
            public static long Count(long[] array, long x, int n)
            {
                long total = 0;
                for (int i = 0; i < n; ++i)
                {
                    total += (array[i] == x ? 1 : 0);
                }
                return total;
            }
            public static long[] ASum(long[] array)
            {
                var res = new long[array.Length + 1];
                for (int i = 0; i < array.Length; ++i)
                {
                    res[i + 1] = res[i] + array[i];
                }
                return res;
            }
            public static long[] ASum(long[] array, long P)
            {
                var res = new long[array.Length + 1];
                for (int i = 0; i < array.Length; ++i)
                {
                    res[i + 1] = (res[i] + array[i]) % P;
                }
                return res;
            }

            public static long GCD(long a, long b)
            {
                while (b > 0)
                {
                    var c = a % b;
                    a = b;
                    b = c;
                }
                return a;
            }
            public static bool IsPrime(long x)
            {
                var fact = Factorize(x);
                return fact.Count == 1 && fact.Values.ElementAt(0) == 1;
            }
            public static Dictionary<long,int> Factorize(long x)
            {
                var dict = new Dictionary<long, int>();
                long minp = 2;
                while (x > 1)
                {
                    bool ok = false;
                    for(long p = minp; p * p <= x; ++p)
                    {
                        if (x % p == 0)
                        {
                            if (!dict.ContainsKey(p))
                            {
                                dict.Add(p, 0);
                            }
                            dict[p]++;
                            minp = p;
                            x /= p;
                            ok = true;
                            break;
                        }
                    }
                    if (!ok)
                    {
                        if (!dict.ContainsKey(x)) { dict.Add(x, 0); }
                        dict[x]++;
                        break;
                    }
                }
                return dict;
            }
            public static long Inverse(long a, long p)
            {
                long b = p, c = 1, d = 0, e = 0, f = 0;
                while (b != 0)
                {
                    e = a / b;
                    f = b; b = a - e * b; a = f;
                    f = d; d = c - e * d; c = f;
                }
                c %= p;
                return c + (c < 0 ? p : 0);
            }
            public static long[] Facts(int n, long p)
            {
                var facts = new long[n + 1];
                facts[0] = 1;
                for (int i = 1; i <= n; ++i)
                {
                    facts[i] = (i * facts[i - 1]) % p;
                }
                return facts;
            }
            public static Func<int,int,long> Combin(int n, long P)
            {
                var fact = Facts(n, P);
                var inv = fact.Select(x => Inverse(x, P)).ToArray();
                return (N, M) =>
                {
                    if (N < 0 || N < M || M < 0) { return 0; }
                    return (((fact[N] * inv[M]) % P) * inv[N - M]) % P;
                };
            }

            /// <summary>
            /// Calc Combin(n,m) where n is large and m<=mMax
            /// </summary>
            public static Func<int,int,long> CombinLarge(int mMax, long p)
            {
                var inv = Enumerable.Range(0, mMax + 1).Select(x => Util.Inverse(x, Util.P)).ToArray();
                return (n, m) =>
                {
                    long mult = 1;
                    for (int i = 1; i <= m; ++i)
                    {
                        mult = (((mult * (n + 1 - i)) % Util.P) * inv[i]) % Util.P;
                    }
                    return mult;
                };
            }

            /// <summary>
            /// Calc 2^n mod p
            /// </summary>
            public static long Power2(long n, long p)
            {
                return Power(2, n, p);
            }
            public static long Power(long a, long n, long p)
            {
                long res = 1;
                long mult = a;
                for (int i = 0; i < 40; ++i)
                {
                    if ((n & ((long)1 << i)) > 0)
                    {
                        res = (res * mult) % p;
                    }
                    mult = (mult * mult) % p;
                }
                return res;
            }

            public static int Lowerbound<T1,T2>(SortedList<T1,T2> list, T1 x) where T1 : IComparable<T1>
            {
                return Lowerbound(list.Keys, x);
            }
            public static int Upperbound<T1, T2>(SortedList<T1, T2> list, T1 x) where T1 : IComparable<T1>
            {
                return Upperbound(list.Keys, x);
            }
            public static int Lowerbound<T>(IList<T> list, T x) where T : IComparable<T>
            {
                int min = 0;
                int max = list.Count;
                while (min < max)
                {
                    var mid = (min + max) / 2;
                    if (list[mid].CompareTo(x) >= 0)
                    {
                        max = mid;
                    }
                    else
                    {
                        min = mid + 1;
                    }
                }
                return max;
            }
            public static int Upperbound<T>(IList<T> list, T x) where T : IComparable<T>
            {
                int min = 0;
                int max = list.Count;
                while (min < max)
                {
                    var mid = (min + max) / 2;
                    if (list[mid].CompareTo(x) > 0)
                    {
                        max = mid;
                    }
                    else
                    {
                        min = mid + 1;
                    }
                }
                return max;
            }
            /// <summary>
            /// Thanks to yambe2002-san
            /// </summary>
            public class BST<T> where T : IComparable
            {
                public class Node
                {
                    public T V;
                    public Node L, R;
                    public int Count;
                    public Node(T v) { V = v; Count = 1; }
                }
                static Random _rnd = new Random();
                public static int Count(Node t) { return t == null ? 0 : t.Count; }
                static Node Update(Node t) { t.Count = Count(t.L) + Count(t.R) + 1; return t; }
                public static Node Merge(Node l, Node r)
                {
                    if (l == null || r == null) return l == null ? r : l;
                    if ((double)Count(l) / (double)(Count(l) + Count(r)) > _rnd.NextDouble())
                    { l.R = Merge(l.R, r); return Update(l); }
                    else { r.L = Merge(l, r.L); return Update(r); }
                }
                public static Tuple<Node, Node> Split(Node t, int k)
                {
                    if (t == null) return new Tuple<Node, Node>(null, null);
                    if (k <= Count(t.L))
                    {
                        var s = Split(t.L, k);
                        t.L = s.Item2;
                        return new Tuple<Node, Node>(s.Item1, Update(t));
                    }
                    else
                    {
                        var s = Split(t.R, k - Count(t.L) - 1);
                        t.R = s.Item1;
                        return new Tuple<Node, Node>(Update(t), s.Item2);
                    }
                }
                public static Node Remove(Node t, T v)
                {
                    if (Find(t, v) == null) return t;
                    return RemoveAt(t, LowerBound(t, v));
                }
                public static Node RemoveAt(Node t, int k)
                {
                    var s = Split(t, k);
                    var s2 = Split(s.Item2, 1);
                    return Merge(s.Item1, s2.Item2);
                }
                public static bool Contains(Node t, T v) { return Find(t, v) != null; }
                public static Node Find(Node t, T v)
                {
                    while (t != null)
                    {
                        var cmp = t.V.CompareTo(v);
                        if (cmp > 0) t = t.L;
                        else if (cmp < 0) t = t.R;
                        else break;
                    }
                    return t;
                }

                public static Node FindByIndex(Node t, int idx)
                {
                    if (t == null) return null;
                    var currentIdx = Count(t) - Count(t.R) - 1;
                    while (t != null)
                    {
                        if (currentIdx == idx) return t;
                        if (currentIdx > idx)
                        { t = t.L; currentIdx -= (Count(t == null ? null : t.R) + 1); }
                        else { t = t.R; currentIdx += (Count(t == null ? null : t.L) + 1); }
                    }
                    return null;
                }
                public static int UpperBound(Node t, T v)
                {
                    var torg = t;
                    if (t == null) return -1;

                    var ret = Int32.MaxValue;
                    var idx = Count(t) - Count(t.R) - 1;
                    while (t != null)
                    {
                        var cmp = t.V.CompareTo(v);

                        if (cmp > 0)
                        {
                            ret = Math.Min(ret, idx);
                            t = t.L;
                            idx -= (Count(t == null ? null : t.R) + 1);
                        }
                        else if (cmp <= 0)
                        {
                            t = t.R;
                            idx += (Count(t == null ? null : t.L) + 1);
                        }
                    }
                    return ret == Int32.MaxValue ? Count(torg) : ret;
                }
                public static int LowerBound(Node t, T v)
                {
                    var torg = t;
                    if (t == null) return -1;

                    var idx = Count(t) - Count(t.R) - 1;
                    var ret = Int32.MaxValue;
                    while (t != null)
                    {
                        var cmp = t.V.CompareTo(v);
                        if (cmp >= 0)
                        {
                            if (cmp == 0) ret = Math.Min(ret, idx);
                            t = t.L;
                            if (t == null) ret = Math.Min(ret, idx);
                            idx -= t == null ? 0 : (Count(t.R) + 1);
                        }
                        else if (cmp < 0)
                        {
                            t = t.R;
                            idx += (Count(t == null ? null : t.L) + 1);
                            if (t == null) return idx;
                        }
                    }
                    return ret == Int32.MaxValue ? Count(torg) : ret;
                }
                public static Node Insert(Node t, T v)
                {
                    var ub = LowerBound(t, v);
                    return InsertByIdx(t, ub, v);
                }
                static Node InsertByIdx(Node t, int k, T v)
                {
                    var s = Split(t, k);
                    return Merge(Merge(s.Item1, new Node(v)), s.Item2);
                }
                public static IEnumerable<T> Enumerate(Node t)
                {
                    var ret = new List<T>();
                    Enumerate(t, ret);
                    return ret;
                }
                static void Enumerate(Node t, List<T> ret)
                {
                    if (t == null) return;
                    Enumerate(t.L, ret);
                    ret.Add(t.V);
                    Enumerate(t.R, ret);
                }
            }
            public class Set<T> where T : IComparable
            {
                protected BST<T>.Node _root;
                public T this[int idx] { get { return ElementAt(idx); } }
                public int Count() { return BST<T>.Count(_root); }
                public virtual void Insert(T v)
                {
                    if (_root == null) _root = new BST<T>.Node(v);
                    else
                    {
                        if (BST<T>.Find(_root, v) != null) return;
                        _root = BST<T>.Insert(_root, v);
                    }
                }
                public void Clear() { _root = null; }
                public void Remove(T v) { _root = BST<T>.Remove(_root, v); }
                public bool Contains(T v) { return BST<T>.Contains(_root, v); }
                public T ElementAt(int k)
                {
                    var node = BST<T>.FindByIndex(_root, k);
                    if (node == null) throw new IndexOutOfRangeException();
                    return node.V;
                }
                public int Count(T v) { return BST<T>.UpperBound(_root, v) - BST<T>.LowerBound(_root, v); }
                public int LowerBound(T v) { return BST<T>.LowerBound(_root, v); }
                public int UpperBound(T v) { return BST<T>.UpperBound(_root, v); }
                public Tuple<int, int> EqualRange(T v)
                {
                    if (!Contains(v)) return new Tuple<int, int>(-1, -1);
                    return new Tuple<int, int>(BST<T>.LowerBound(_root, v), BST<T>.UpperBound(_root, v) - 1);
                }
                public List<T> ToList() { return new List<T>(BST<T>.Enumerate(_root)); }
            }
            public class MultiSet<T> : Set<T> where T : IComparable
            { public override void Insert(T v) { if (_root == null) { _root = new BST<T>.Node(v); } else { _root = BST<T>.Insert(_root, v); } } }
        }

        class Graph
        {
            public static HashSet<int> ConnectedComponent(Dictionary<int, HashSet<int>> adj, int idx)
            {
                HashSet<int> comp = new HashSet<int>();
                comp.Add(idx);
                List<int> target = new List<int>();
                target.Add(idx);
                while (target.Count > 0)
                {
                    List<int> next = new List<int>();
                    foreach (var p in target)
                    {
                        foreach (var f in adj[p])
                        {
                            if (!comp.Contains(f))
                            {
                                next.Add(f);
                                comp.Add(f);
                            }
                        }
                    }
                    target = next;
                }
                return comp;
            }
        }

        static string ReadStr()
        {
            return Console.ReadLine();
        }
        static string[] ReadStrN()
        {
            return Console.ReadLine().Split(' ');
        }
        static int ReadInt()
        {
            return Convert.ToInt32(Console.ReadLine());
        }
        static int[] ReadIntN()
        {
            return Console.ReadLine().Split(' ').Select(s => Convert.ToInt32(s)).ToArray();
        }
        static long ReadLong()
        {
            return Convert.ToInt64(Console.ReadLine());
        }
        static long[] ReadLongN()
        {
            return Console.ReadLine().Split(' ').Select(s => Convert.ToInt64(s)).ToArray();
        }
        static Tuple<string, string>[] ReadTupleStr(int n)
        {
            List<Tuple<string, string>> list = new List<Tuple<string, string>>();
            for (int i = 0; i < n; ++i)
            {
                var ab = ReadStrN();
                list.Add(new Tuple<string, string>(ab[0], ab[1]));
            }
            return list.ToArray();
        }
        static Tuple<long, long>[] ReadTupleLong(int n)
        {
            List<Tuple<long, long>> list = new List<Tuple<long, long>>();
            for (int i = 0; i < n; ++i)
            {
                var ab = ReadLongN();
                list.Add(new Tuple<long, long>(ab[0], ab[1]));
            }
            return list.ToArray();
        }
        static Tuple<double, double>[] ReadTupleDouble(int n)
        {
            List<Tuple<double, double>> list = new List<Tuple<double, double>>();
            for (int i = 0; i < n; ++i)
            {
                var ab = ReadLongN();
                list.Add(new Tuple<double, double>(ab[0], ab[1]));
            }
            return list.ToArray();
        }
        static Dictionary<int, HashSet<int>> ReadAdjacencyList(int N, int M)
        {
            Dictionary<int, HashSet<int>> dict = new Dictionary<int, HashSet<int>>();
            for (int i = 0; i <= N; ++i)
            {
                dict.Add(i, new HashSet<int>());
            }
            for (int i = 0; i < M; ++i)
            {
                var ab = ReadIntN();
                dict[ab[0]].Add(ab[1]);
                dict[ab[1]].Add(ab[0]);
            }
            return dict;
        }
    }
}