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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)


import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.Scanner;

public class Main {
    static long MOD = 1000000007;

    static public void main(String[] args) {
        Scanner sc = new Scanner(System.in);

        int n = sc.nextInt();
        long[][] arrs = new long[n][n];
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                arrs[i][j] = sc.nextLong();
            }
        }

        List<Vertex> vertexs = new ArrayList<>();
        List<Edge> edges = new ArrayList<>();
        for (int i = 0; i < n; i++) {
            vertexs.add(new Vertex(i));
        }
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                long d = arrs[i][j];
                if (d > 0) {
                    edges.add(new Edge(String.valueOf(i), vertexs.get(i), vertexs.get(j), d));
                }
            }
        }

        Graph graph = new Graph(vertexs, edges);
        WarshallFloydAlgorithm warshallFloydAlgorithm = new WarshallFloydAlgorithm(graph);
        long[][] warshallResutls = warshallFloydAlgorithm.execute();

        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                if (i != j && warshallResutls[i][j] != arrs[i][j]) {
                    System.out.println(-1);
                    System.exit(0);
                }
            }
        }

        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                long longest = Long.MAX_VALUE;
                for (int k = 0; k < n; k++) {
                    long distance = warshallResutls[i][k] + warshallResutls[k][j];
                    if (i != k && j != k && distance < longest) {
                        longest = distance;
                    }
                }

                if (longest == warshallResutls[i][j]) {
                    warshallResutls[i][j] = 1000000000L;
                }
            }
        }

        long ans = 0;
        for (int i = 0; i < n; i++) {
            for (int j = i + 1; j < n; j++) {
                if (warshallResutls[i][j] != 1000000000L) {
                    ans += warshallResutls[i][j];
                }
            }
        }

        for (int i = 0; i < n; i++) {
            System.out.println(Arrays.toString(warshallResutls[i]));

        }

        System.out.println(ans);
    }

    static private class WarshallFloydAlgorithm {

        private final List<Vertex> vertexes;
        private final List<Edge> edges;
        private final long INF = 1000000000L;

        public WarshallFloydAlgorithm(Graph graph) {
            this.vertexes = new ArrayList<>(graph.getVertices());
            this.edges = new ArrayList<>(graph.getEdges());
        }

        public long[][] execute() {
            int n = vertexes.size();

            long[][] distances = new long[n][n];
            for (long[] row : distances)
                Arrays.fill(row, INF);

            for (Edge edge : this.edges) {
                distances[edge.getSource().getId()][edge.getDestination().getId()] = edge.getWeight();
                distances[edge.getDestination().getId()][edge.getSource().getId()] = edge.getWeight();
            }

            for (int k = 0; k < n; k++) {
                for (int i = 0; i < n; i++) {
                    for (int j = 0; j < n; j++) {
                        long d = distances[i][k] + distances[k][j];
                        if (d < distances[i][j])
                            distances[i][j] = d;
                    }
                }
            }

            return distances;
        }
    }

    static private class Graph {
        private final List<Vertex> vertices;
        private final List<Edge> edges;

        public Graph(List<Vertex> vertices, List<Edge> edges) {
            this.vertices = vertices;
            this.edges = edges;
        }


        public List<Vertex> getVertices() {
            return this.vertices;
        }

        public List<Edge> getEdges() {
            return this.edges;
        }
    }

    static private class Edge {
        private final String id;
        private final Vertex source;
        private final Vertex destination;
        private final long weight;

        public Edge(String id, Vertex source, Vertex destination, long weight) {
            this.id = id;
            this.source = source;
            this.destination = destination;
            this.weight = weight;
        }

        public String getId() {
            return this.id;
        }

        public Vertex getDestination() {
            return this.destination;
        }

        public Vertex getSource() {
            return this.source;
        }

        public long getWeight() {
            return this.weight;
        }

        @Override
        public String toString() {
            return "weight: " + this.weight + ", " + this.source.toString() + "," + this.destination.toString();
        }
    }

    static private class Vertex {
        final private int id;

        public Vertex(int id) {
            this.id = id;
        }

        public int getId() {
            return this.id;
        }

        @Override
        public int hashCode() {
            return this.id;
        }

        @Override
        public boolean equals(Object obj) {
            if (this == obj) return true;
            if (obj == null) return false;
            if (getClass() != obj.getClass()) return false;
            Vertex other = (Vertex) obj;
            return this.getId() == other.getId();
        }

        @Override
        public String toString() {
            return "id: " + this.id;
        }
    }
}