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;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using static System.Console;
using static System.Math;
 
public class Solve{
    static public int mod = 1000000007;
    public static void Main(){
        // 方針
        // ワーシャルフロイド
        // 短いほうからn-1個取り出してワーシャルフロイド
        // 以降表と比較して
        var n = rint();
        long[,] tmap = new long[n,n];
        for(int i=0;i<n;i++){
            var temp = longa();
            for(int j=0;j<n;j++){
                tmap[i,j] = temp[j];
            }
        }
        var anst = true;
        long[] map = new long[n*n];
        var num = new int[n*n];
        for(int i=0;i<n*n;i++){
            map[i] = tmap[i/n,i%n];
            num[i] = i;
        }
        Array.Sort(map,num);
        long[,] wf = new long[n,n];
        for(int i=0;i<n;i++){
            for(int j=0;j<n;j++){
                wf[i,j] = 9999999999999;
                if(i==j) wf[i,j] = 0;
            }
        }
        long ans = 0;
        int count =0;
        for(int i=0;i<n*n;i++){
            if(map[i] == 0) continue;
            int tate = num[i]/n;
            int yoko = num[i]%n;
            //WriteLine($"{wf[tate,yoko]} {map[i]} {tate} {yoko}");
            if(wf[tate,yoko] > map[i]){
                wf[tate,yoko] = map[i];
                wf[yoko,tate] = map[i];
                ans += map[i];
                for(int q=0;q<n;q++){
                    for(int w=0;w<n;w++){
                        for(int e=0;e<n;e++){
                            wf[w,e] = Min(wf[w,e],wf[w,q]+wf[q,e]);
                        }
                    }
                }
            }else if(wf[tate,yoko] == map[i]){
                continue;
            }else{
                anst = false;
                break;
            }
        }
        //WriteLine(anst);
        for(int i=0;i<n;i++){
            for(int j=0;j<n;j++){
                if(wf[i,j] != tmap[i,j]) anst = false;
            }
        }
        //WriteArray(wf,n,n);
        //WriteArray(tmap,n,n);
        if(anst){
            WriteLine(ans);
        }else{
            WriteLine(-1);
        }
        
        
        
        
        
        
        
        
        
        
        
    }
    // 1からaまでの数字並べ替え全通り返すやつ
    static List<string> permutation(int a){var list = new List<string>();if(a == 1){list.Add("1");return list;}else{foreach(var k in permutation(a-1)){var ch = k.ToCharArray();for(int i=0;i<=ch.Length;i++){string intemp = "";if(i == 0){intemp += a.ToString();}for(int j=0;j<ch.Length;j++){intemp += ch[j].ToString();if(i == j+1){intemp += a.ToString();}}list.Add(intemp);}}}return list;}
    public static void swap(ref int a,ref int b){int temp = a;a= b;b = temp;}
    static void charswap(ref char a,ref char b){char temp = a;a= b;b = temp;}
    static int ncr(int n,int r){r = Min(r,n-r);long nn = 1;for(int i=n-r+1;i<=n;i++){nn = nn*i%mod;}long rr = 1;for(int i=1;i<=r;i++){rr = rr*i%mod;}rr = square((int)rr,mod-2);nn = nn * rr %mod;return (int)nn;}
    // a^b mod
    static int square(int a,int b){string binary = Convert.ToString(b,2);int bileng = binary.Length;long a_power = a;long value = 1;for(int i=bileng-1;i>=0;i--){if(binary[i] == '1'){value = value*a_power%mod;}a_power = a_power*a_power%mod;}return (int)value;}
    static int square2(int a,int b){long output = 1;var list = new List<long>();int sh = 1;long n = a;list.Add(a);while(sh < b){sh *= 2;n = n*n%mod;list.Add(n);}for(int i=list.Count-1;i>=0;i--){if(b > sh){b -= sh;sh /= 2;output = output*list[i]%mod;}}return (int)output;}
    //各種読取
    static string rstr(){ return ReadLine(); }
    static int rint(){ return int.Parse(ReadLine()); }
    static long rlong(){ return long.Parse(ReadLine()); }
    static string[] stra(){ return ReadLine().Split(' '); }
    static char[] chara(){ string[] a=stra();string b="";for(int i=0;i<a.Length;i++){b+=a[i];}return b.ToCharArray();}
    static int[] inta(){ string[] read_str_array = ReadLine().Split(' '); int[] return_int_array = new int[read_str_array.Length]; for(int countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_int_array[countup_i] = int.Parse(read_str_array[countup_i]); } return return_int_array; }
    static int[,] inta2(int num_array,int in_array){ int[,] int_array2 = new int[num_array,in_array]; for(int i=0;i<num_array;i++){ int[] temp_array = inta(); for(int j=0;j<in_array;j++){ int_array2[i,j] = temp_array[j]; } } return int_array2; }
    static long[] longa(){ string[] read_str_array = ReadLine().Split(' '); long[] return_long_array = new long[read_str_array.Length]; for(long countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_long_array[countup_i] = long.Parse(read_str_array[countup_i]); } return return_long_array; }
    static double[] doublea(){ string[] read_str_array = ReadLine().Split(' '); double[] return_double_array = new double[read_str_array.Length]; for(long countup_i=0;countup_i<read_str_array.Length;countup_i++){ return_double_array[countup_i] = long.Parse(read_str_array[countup_i]); } return return_double_array; }
    // -----------------------------
    static long divideup(long divided,long divid){ long temp_divide = divided/divid; if(divided % divid > 0){ temp_divide++; } return temp_divide; }
    static long GCD(long a,long b){ if(a < b){ long temp = a; a = b; b = temp; } if(a % b == 0){ return b; } else{ long temp = b; b = a%b; a = temp; return GCD(a,b); } }
    static long LCM(long a,long b){ return a * b / GCD(a,b); }
    static void WriteArray(long[,] a,int b,int c){
        for(int i=0;i<b;i++){
            for(int j=0;j<c;j++){
                if(j!=0) Write(" ");
                Write(a[i,j]);
            }
            WriteLine();
        }
    }
}