Problem:
Score : 500 points

Problem Statement

N players will participate in a tennis tournament. We will call them Player 1, Player 2, \ldots, Player N.

The tournament is round-robin format, and there will be N(N-1)/2 matches in total.
Is it possible to schedule these matches so that all of the following conditions are satisfied? If the answer is yes, also find the minimum number of days required.

Each player plays at most one matches in a day.

Each player i (1 \leq i \leq N) plays one match against Player A_{i, 1}, A_{i, 2}, \ldots, A_{i, N-1} in this order.

Constraints

3 \leq N \leq 1000

1 \leq A_{i, j} \leq N

A_{i, j} \neq i

A_{i, 1}, A_{i, 2}, \ldots, A_{i, N-1} are all different.

Input

Input is given from Standard Input in the following format:

N
A_{1, 1} A_{1, 2} \ldots A_{1, N-1}
A_{2, 1} A_{2, 2} \ldots A_{2, N-1}
:
A_{N, 1} A_{N, 2} \ldots A_{N, N-1}

Output

If it is possible to schedule all the matches so that all of the conditions are satisfied, print the minimum number of days required; if it is impossible, print -1.

Sample Input 1

3
2 3
1 3
1 2

Sample Output 1

3

All the conditions can be satisfied if the matches are scheduled for three days as follows:

Day 1: Player 1 vs Player 2

Day 2: Player 1 vs Player 3

Day 3: Player 2 vs Player 3

This is the minimum number of days required.

Sample Input 2

4
2 3 4
1 3 4
4 1 2
3 1 2

Sample Output 2

4

All the conditions can be satisfied if the matches are scheduled for four days as follows:

Day 1: Player 1 vs Player 2, Player 3 vs Player 4

Day 2: Player 1 vs Player 3

Day 3: Player 1 vs Player 4, Player 2 vs Player 3

Day 4: Player 2 vs Player 4

This is the minimum number of days required.

Sample Input 3

3
2 3
3 1
1 2

Sample Output 3

-1

Any scheduling of the matches violates some condition.

Solution in C# (.NET Core 3.1.201) (judge verdict: Time Limit Exceeded)

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

class MainClass : Scanner
{
    static void Main()
    {
        int N = RInt();
        var G = new Graph<int>(N*(N-1)/2);
        var G1 = new Graph<int>(N * (N - 1) / 2);
        var A = new int[N][];
        for (int i = 0; i < N; i++)
        {
            A[i] = RInts(_=>int.Parse(_)-1);
        }
        int cnt = 0;
        var pairtoindex = new Dictionary<(int, int), int>();
        for (int i = 0; i < N; i++)
        {
            for (int j = i+1; j < N; j++)
            {
                pairtoindex[(i, j)] = cnt++;
            }
        }
        for (int i = 0; i < N; i++)
        {
            for (int j = 0; j < N-2; j++)
            {
                int from = -1, to = -1;
                if (i > A[i][j]) from = pairtoindex[(A[i][j], i)];
                else from = pairtoindex[(i, A[i][j])];
                if (i > A[i][j+1]) to = pairtoindex[(A[i][j+1], i)];
                else to = pairtoindex[(i, A[i][j+1])];
                G.Add(from,to);
                G1.Add(to, from);
            }
        }
        
        var sp = new List<int>();

        for (int i = 0; i < N*(N-1)/2; i++)
        {
            int degree = G[i].Count;
            if (degree == 0)
            {
                sp.Add(i);
            }
        }
        if (sp.Count == 0)
        {
            WriteLine(-1);return;
        }
        int ans = 0;
        foreach (var i in sp)
        {
            var days = new int[N * (N - 1)];
            var que = new Queue<(int, int)>();
            que.Enqueue((i, 1));
            days[i] = 1;
            while (que.Count > 0)
            {
                var v = que.Dequeue();
                int now = v.Item1;
                int day = v.Item2;
                if (day > N * (N - 1) / 2)
                {
                    WriteLine(-1);
                    continue;
                }
                foreach (var j in G1[now])
                {
                    var nday = day + 1;
                    if (days[j] < nday)
                    {
                        days[j] = nday;
                        que.Enqueue((j, nday));
                    }
                }
            }
            ans = Max(ans, days.Max());
        }
        WriteLine(ans);
    }

}
class Graph<T>
{
    private SortedSet<T>[] G;
    public Graph(int size)
    {
        G = new SortedSet<T>[size];
        for (int i = 0; i < size; i++)
        {
            G[i] = new SortedSet<T>();
        }
    }

    public SortedSet<T> this[int i] => G[i];
    public void Add(int from, T to) => G[from].Add(to);
    public SortedSet<T>[] ToListArray() => G;
    public int Length => G.Length;
}

class Scanner
{
    public static int RInt() => int.Parse(Console.ReadLine());
    public static long RLong() => long.Parse(Console.ReadLine());
    public static string RString() => Console.ReadLine();
    public static double RDouble() => double.Parse(Console.ReadLine());

    public static int[] RInts() => Array.ConvertAll(Console.ReadLine().Split(), int.Parse);
    public static int[] RInts(Func<string, int> func) => Console.ReadLine().Split().Select(func).ToArray();
    public static long[] RLongs() => Array.ConvertAll(Console.ReadLine().Split(), long.Parse);
    public static long[] RLongs(Func<string, long> func) => Console.ReadLine().Split().Select(func).ToArray();
    public static double[] RDoubles() => Array.ConvertAll(Console.ReadLine().Split(), double.Parse);
    public static double[] RDoubles(Func<string, double> func) => Console.ReadLine().Split().Select(func).ToArray();
    public static string[] RStrings() => Console.ReadLine().Split();
    //public static string[] RStrings(Func<string, string> func) => Console.ReadLine().Split().Select(func).ToArray();
    public static T1 ReadStream<T1>()
    {
        var r = RStrings();
        var r1 = (T1)Convert.ChangeType(r[0], typeof(T1));
        return r1;
    }
    public static (T1, T2) ReadStream<T1, T2>()
    {
        var r = RStrings();
        var r1 = (T1)Convert.ChangeType(r[0], typeof(T1));
        var r2 = (T2)Convert.ChangeType(r[1], typeof(T2));
        return (r1, r2);
    }
    public static (T1, T2, T3) ReadStream<T1, T2, T3>()
    {
        var r = RStrings();
        var r1 = (T1)Convert.ChangeType(r[0], typeof(T1));
        var r2 = (T2)Convert.ChangeType(r[1], typeof(T2));
        var r3 = (T3)Convert.ChangeType(r[2], typeof(T3));
        return (r1, r2, r3);
    }
    public static (T1, T2, T3, T4) ReadStream<T1, T2, T3, T4>()
    {
        var r = RStrings();
        var r1 = (T1)Convert.ChangeType(r[0], typeof(T1));
        var r2 = (T2)Convert.ChangeType(r[1], typeof(T2));
        var r3 = (T3)Convert.ChangeType(r[2], typeof(T3));
        var r4 = (T4)Convert.ChangeType(r[3], typeof(T4));
        return (r1, r2, r3, r4);
    }
}