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.Linq;
using System.Text;

class TEST{
	static void Main(){
		Sol mySol =new Sol();
		mySol.Solve();
	}
}

class Sol{
	public void Solve(){
		
		long[][] WF = new long[N][];
		for(int i=0;i<N;i++){
			WF[i] = (long[]) A[i].Clone();
		}
		
		for(int k=0;k<N;k++){
			for(int i=0;i<N;i++){
				for(int j=0;j<N;j++){
					WF[i][j] = Math.Min(WF[i][j],WF[i][k] + WF[k][j]);
				}
			}
		}
		
		for(int i=0;i<N;i++){
			for(int j=0;j<N;j++){
				if(WF[i][j] != A[i][j]){
					Console.WriteLine(-1);
					return;
				}
			}
		}
		
		List<Pair> L = new List<Pair>();
		for(int i=0;i<N;i++){
			for(int j=i+1;j<N;j++){
				L.Add(new Pair(i*1000+j,A[i][j]));
			}
		}
		L.Sort((p,q) => p.Dist.CompareTo(q.Dist));
		
		long Inf = (long)1e18;
		for(int i=0;i<N;i++){
			for(int j=0;j<N;j++){
				WF[i][j] = i == j ? 0 : Inf;
			}
		}
		
		long tot = 0;
		foreach(var p in L){
			int f = (int) p.FT/1000;
			int t = (int) p.FT%1000;
			if(WF[f][t] > A[f][t]){
				WF[f][t] = WF[t][f] = p.Dist;
				tot += p.Dist;
				for(int i=0;i<N;i++){
					for(int j=0;j<N;j++){
						WF[i][j] = Math.Min(WF[i][j],WF[i][f]+WF[f][t]+WF[t][j]);
						WF[i][j] = Math.Min(WF[i][j],WF[i][t]+WF[t][f]+WF[f][j]);
					}
				}
			}
		}
		
		Console.WriteLine(tot);
		
	}
	
	class Pair{
		public long FT;
		public long Dist;
		public Pair(long ft, long dist){
			FT = ft; Dist = dist;
		}
	}
	
	
	int N;
	long[][] A;
	public Sol(){
		N = ri();
		A = new long[N][];
		for(int i=0;i<N;i++) A[i] = rla();
	}

	static String rs(){return Console.ReadLine();}
	static int ri(){return int.Parse(Console.ReadLine());}
	static long rl(){return long.Parse(Console.ReadLine());}
	static double rd(){return double.Parse(Console.ReadLine());}
	static String[] rsa(char sep=' '){return Console.ReadLine().Split(sep);}
	static int[] ria(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>int.Parse(e));}
	static long[] rla(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>long.Parse(e));}
	static double[] rda(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>double.Parse(e));}
}


class UnionFind{
	
	int[] parent;
	int[] mem;
	int compo;
	int N;
	
	public UnionFind(UnionFind v){
		// copy constructor
		N = v.N;
		parent = new int[N];
		mem = new int[N];
		for(int i=0;i<N;i++){
			parent[i] = v.parent[i];
			mem[i] = v.mem[i];
		}
		compo = v.compo;
	}
	public UnionFind(int n_){
		Initialize(n_);
	}
	
	public void Initialize(int n_){
		N = n_;
		parent = new int[N];
		mem=new int[N];
		for(int i=0;i<N;i++){
			parent[i] = i;
			mem[i] = 1;
		}
		compo = N;
	}
	
	public int Parent(int a){
		if(parent[a] == a)return a;
		return parent[a] = Parent(parent[a]);
	}
	
	public bool IsUnited(int a,int b){
		return Parent(a) == Parent(b);
	}
	
	public void Unite(int a,int b){
		a = Parent(a); b = Parent(b);//Parent()を呼ぶことでa以上のノードは全てparentがrootになる
		if(a == b)return;
		parent[a] = b;
		mem[b] += mem[a];
		compo -= 1;
	}
	
	public bool IsRoot(int x){
		return x == parent[x];
	}
	public int MemCnt(int x){
		return mem[Parent(x)];
	}
	
	public void Dump(){
		for(int i=0;i<parent.Length;i++){
			Console.Write(i == 0 ? "{0}" : " {0}",parent[i]);
		}
		Console.WriteLine("");
	}
	
	public int Compo{
		get{
			return compo;
		}
	}
	
}