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: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Text;
using System.Linq;

namespace AtTest.D_Challenge
{
    class ABC_074
    {
        static void Main(string[] args)
        {
            Method(args);
            Console.ReadLine();
        }

        static void Method(string[] args)
        {
            int n = ReadInt();
            var distances = new long[n * (n - 1) / 2][];
            int itr = 0;
            var nodes = new Node[n];
            for (int i = 0; i < n; i++)
            {
                long[] dist = ReadLongs();
                for (int j = i + 1; j < n; j++)
                {
                    distances[itr] = new long[3] { dist[j], i, j };
                    itr++;
                }
                nodes[i] = new Node();
            }
            distances = distances.OrderBy(a => a[0]).ToArray();
            /*
            for (int i = 0; i < distances.Length; i++)
            {
                Console.WriteLine(
                distances[i][0] + " " + distances[i][1] + " " + distances[i][2]);
            }
            */
            long res = 0;
            //Console.WriteLine(distances.Length);
            for(int i = 0; i < distances.Length; i++)
            {
                long distance = GetDistance((int)distances[i][1],
                    (int)distances[i][2], nodes);
                //Console.WriteLine(distance);
                if (distance > distances[i][0])
                {
                    nodes[distances[i][1]].edges.Add(
                        new Edge((int)distances[i][2], distances[i][0]));
                    nodes[distances[i][2]].edges.Add(
                        new Edge((int)distances[i][1], distances[i][0]));
                    res += distances[i][0];
                }
                else if (distance < distances[i][0])
                {
                    Console.WriteLine(-1);
                    return;
                }
            }
            Console.WriteLine(res);
        }

        static long GetDistance(int start, int goal, Node[] nodes)
        {
            var posQueue = new Queue<int>();
            var dists = new long[nodes.Length];
            for(int i = 0; i < dists.Length; i++)
            {
                dists[i] = long.MaxValue;
            }
            posQueue.Enqueue(start);
            dists[start] = 0;
            while (posQueue.Count > 0)
            {
                int now = posQueue.Dequeue();
                for(int i = 0; i < nodes[now].edges.Count; i++)
                {
                    long d = dists[now] + nodes[now].edges[i].distance;
                    if (d < dists[nodes[now].edges[i].toIndex])
                    {
                        dists[nodes[now].edges[i].toIndex] = d;
                        posQueue.Enqueue(nodes[now].edges[i].toIndex);
                        //Console.WriteLine("Set " +i+ " "+ + d);
                    }
                }
            }
            return dists[goal];
        }

        class Node
        {
            public List<Edge> edges;
            public Node()
            {
                edges = new List<Edge>();
            }
        }

        class Edge
        {
            public int toIndex;
            public long distance;

            public Edge(int toIndex, long distance)
            {
                this.toIndex = toIndex;
                this.distance = distance;
            }
        }
        private static string Read() { return Console.ReadLine(); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
    }
}