Problem:
Score : 500 points

Problem Statement

In Takahashi Kingdom, which once existed, there are N cities, and some pairs of cities are connected bidirectionally by roads.
The following are known about the road network:

People traveled between cities only through roads. It was possible to reach any city from any other city, via intermediate cities if necessary.

Different roads may have had different lengths, but all the lengths were positive integers.

Snuke the archeologist found a table with N rows and N columns, A, in the ruin of Takahashi Kingdom.
He thought that it represented the shortest distances between the cities along the roads in the kingdom.

Determine whether there exists a road network such that for each u and v, the integer A_{u, v} at the u-th row and v-th column of A is equal to the length of the shortest path from City u to City v.
If such a network exist, find the shortest possible total length of the roads.

Constraints

1 \leq N \leq 300

If i ≠ j, 1 \leq A_{i, j} = A_{j, i} \leq 10^9.

A_{i, i} = 0

Inputs

Input is given from Standard Input in the following format:

N
A_{1, 1} A_{1, 2} ... A_{1, N}
A_{2, 1} A_{2, 2} ... A_{2, N}
...
A_{N, 1} A_{N, 2} ... A_{N, N}

Outputs

If there exists no network that satisfies the condition, print -1.
If it exists, print the shortest possible total length of the roads.

Sample Input 1

3
0 1 3
1 0 2
3 2 0

Sample Output 1

3

The network below satisfies the condition:

City 1 and City 2 is connected by a road of length 1.

City 2 and City 3 is connected by a road of length 2.

City 3 and City 1 is not connected by a road.

Sample Input 2

3
0 1 3
1 0 1
3 1 0

Sample Output 2

-1

As there is a path of length 1 from City 1 to City 2 and City 2 to City 3, there is a path of length 2 from City 1 to City 3.
However, according to the table, the shortest distance between City 1 and City 3 must be 3.

Thus, we conclude that there exists no network that satisfies the condition.

Sample Input 3

5
0 21 18 11 28
21 0 13 10 26
18 13 0 23 13
11 10 23 0 17
28 26 13 17 0

Sample Output 3

82

Sample Input 4

3
0 1000000000 1000000000
1000000000 0 1000000000
1000000000 1000000000 0

Sample Output 4

3000000000

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Collections.ObjectModel;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

public static class TemplateExtension
{
    public static X[] MakeArray<X>(this int count, Func<int, X> func)
    {
        var xs = new X[count];
        for (var i = 0; i < count; i++)
        {
            xs[i] = func(i);
        }
        return xs;
    }

    public static int[] Range(this int count, int start = 0)
    {
        return count.MakeArray(i => i + start);
    }

    public static string Intercalate<X>(this IEnumerable<X> @this, string separator)
    {
        return string.Join(separator, @this);
    }

    public sealed class ValueIndexPair<T>
        : Tuple<T, int>
    {
        public T Value { get { return Item1; } }
        public int Index { get { return Item2; } }

        public ValueIndexPair(T value, int index)
            : base(value, index)
        {
        }
    }

    public static IEnumerable<ValueIndexPair<X>> Indexed<X>(this IEnumerable<X> @this)
    {
        var i = 0;
        foreach (var x in @this)
        {
            yield return new ValueIndexPair<X>(x, i);
            i++;
        }
    }
}

public sealed class Scanner
{
    readonly TextReader reader;
    readonly StringBuilder sb = new StringBuilder();

    /// <summary>
    /// Reads next word separated by spaces.
    /// </summary>
    public string Word()
    {
        sb.Clear();

        while (true)
        {
            var r = reader.Read();

            if (r == '\r')
            {
                if (reader.Peek() == '\n') reader.Read();
                break;
            }
            else if (r == -1 || r == ' ' || r == '\n')
            {
                break;
            }
            else
            {
                sb.Append((char)r);
            }
        }

        return sb.ToString();
    }

    /// <summary>
    /// Reads next word as <see cref="int"/>.
    /// </summary>
    public int N()
    {
        return int.Parse(Word());
    }

    /// <summary>
    /// Reads next word as <see cref="long"/>.
    /// </summary>
    public long L()
    {
        return long.Parse(Word());
    }

    /// <summary>
    /// Reads next word as <see cref="double"/>.
    /// </summary>
    public double F()
    {
        return double.Parse(Word());
    }

    public int[] Ns(int count)
    {
        return count.MakeArray(_ => N());
    }

    public long[] Ls(int count)
    {
        return count.MakeArray(_ => L());
    }

    public double[] Fs(int count)
    {
        return count.MakeArray(_ => F());
    }

    /// <summary>
    /// Reads next line and splits it by spaces.
    /// </summary>
    public X[] Words<X>(Func<string, X> func)
    {
        return reader.ReadLine().Split(' ').Select(func).ToArray();
    }

    public Scanner(TextReader reader)
    {
        this.reader = reader;
    }
}

public partial class Program
{
    readonly TextReader input;
    readonly TextWriter output;
    readonly Scanner scanner;

    void WriteLine(int value)
    {
        output.WriteLine(value);
    }

    void WriteLine(long value)
    {
        output.WriteLine(value);
    }

    void WriteLine(double value)
    {
        output.WriteLine(value);
    }

    void WriteLine(char value)
    {
        output.WriteLine(value);
    }

    void WriteLine(string value)
    {
        output.WriteLine(value);
    }

    public Program(TextReader input, TextWriter output)
    {
        this.input = input;
        this.output = output;
        scanner = new Scanner(input);
    }

    public static void Main(string[] args)
    {
#if DEBUG
        using (var writer = new VainZero.IO.DebugTextWriter(Console.Out))
#else
        var writer = Console.Out;
#endif
        {
            new Program(Console.In, writer).EntryPoint();
        }
    }
}

public sealed class Edge
    : IComparable<Edge>
{
    public int U, V;
    public long W;

    public Edge(int u, int v, long w)
    {
        U = u;
        V = v;
        W = w;
    }

    public int CompareTo(Edge other)
    {
        return Comparer<long>.Default.Compare(W, other.W);
    }
}

public sealed partial class Program
{
    int n;
    long[][] md;

    bool Verify()
    {
        for (var k = 0; k < n; k++)
        {
            for (var u = 0; u < n; u++)
            {
                for (var v = 0; v < n; v++)
                {
                    if (md[u][v] > md[u][k] + md[k][v])
                    {
                        return false;
                    }
                }
            }
        }
        return true;
    }

    long Solve()
    {
        if (!Verify())
        {
            return -1;
        }

        /*
        var es = new Edge[n * n];
        for (var i = 0; i < n; i++)
        {
            for (var j = 0; j < n; j++)
            {
                es[i] = new Edge(i, j, md[i][j]);
            }
        }
        Array.Sort(es);

        var d = n.MakeArray(i => n.MakeArray(j => (long)int.MaxValue));
        for (var i = 0; i < n; i++)
        {
            d[i][i] = 0;
        }

        var notmin = 0;
        for (var i = 0; i < n; i++)
        {
            for (var j = 0; j < n; j++)
            {
                if (d[i][j] != md[i][j]) notmin++;
            }
        }

        for (var ei = 0; ei < es.Length; ei++)
        {
            var u = es[ei].U;
            var v = es[ei].V;
            var w = es[ei].W;

            var bipassExists = false;
            for (var k = 0; k < n; k++)
            {
                if (d[u][k] + d[k][v] == d[u][v])
                {

                }
            }
        }
        */

        var es = new HashSet<long>();
        for (var u = 0; u < n; u++)
        {
            for (var v = u + 1; v < n; v++)
            {
                es.Add(((long)u << 32) + v);
            }
        }

        {
            for (var u = 0; u < n; u++)
            {
                for (var v = 0; v < n; v++)
                {
                    if (u == v) continue;

                    var e = ((long)Math.Min(u, v) << 32) + Math.Max(u, v);
                    if (!es.Contains(e)) continue;

                    var bipassExists = false;
                    for (var k = 0; k < n; k++)
                    {
                        if (k == u || k == v) continue;

                        if (md[u][k] + md[k][v] == md[u][v])
                        {
                            bipassExists = true;
                            break;
                        }
                    }

                    if (bipassExists)
                    {
                        es.Remove(e);
                    }
                }
            }
        }

        return
            es.Sum(e =>
            {
                var u = (int)(e >> 32);
                var v = (int)(e & 0xFFFFFFFF);
                return md[u][v];
            });
    }

    void EntryPoint()
    {
        var q = scanner;
        n = q.N();
        md = n.MakeArray(i => n.MakeArray(j => q.L()));
        WriteLine(Solve());
    }
}