Suppose that the lengths of links in a network G1 V1 E1 wit

Suppose that the lengths of links in a network G_1 = (V_1, E_1) with V_1 = {1, ..., 7} is given by the following matrix A_1 = [a_ij]. A_1 = [0 infinity 4 10 3 infinity infinity infinity 0 -1 -3 2 11 0 infinity 9 0 8 3 2 1 infinity 4 0 0 8 6 3 infinity 0 1 2 0 3 -1 infinity -1 -1 3 2 0 0 infinity 4 3 infinity infinity 2 0] Does the system of Bellman\'s equations have a unique finite solution? Justify your answer. Solve the system of Bellman\'s equations by using the Bellman-Ford method.

Solution

/* Though the question had insufficient information i tried to answer the question .

If this is not what you are looking for please provide with the following information.

1. Do you require the Bellman-Ford algorithm to find the shortest distance from a source?

2.Provide the source vertex.

3. Graph g diagram.

4. Or this is relaed to Maths category , where you want the mathematical evaluation of Bellman equations?

*/

import java.util.Arrays;

public class BellmanFord {
   public static final int MAX_VALUE = 999;

   public static int[] bellmanFordEvaluation(int source, int adjacencymatrix[][]) {
       int noOfvertices = adjacencymatrix.length;
       int distances[] = new int[noOfvertices];

       Arrays.fill(distances, MAX_VALUE);
       distances[source] = 0;
       for (int node = 0; node < noOfvertices - 1; node++) {
           for (int sourcenode = 0; sourcenode < noOfvertices; sourcenode++) {
               for (int destinationnode = 0; destinationnode < noOfvertices; destinationnode++) {
                   if (adjacencymatrix[sourcenode][destinationnode] != MAX_VALUE) {
                       if (distances[destinationnode] > distances[sourcenode]
                               + adjacencymatrix[sourcenode][destinationnode])
                           distances[destinationnode] = distances[sourcenode]
                                   + adjacencymatrix[sourcenode][destinationnode];
                   }
               }
           }
       }

       return distances;
   }

   public static void main(String... arg) {
       int noOfvertices = 7;

       int inf = MAX_VALUE;
       int adjacencymatrix[][] = { { 0, inf, 4, 10, 3, inf, inf }, { inf, 0, -1, -3, 2, 11, 0 },
               { inf, 9, 0, 8, 3, 2, 1 }, { inf, 4, 0, 0, 8, 6, 3 }, { inf, 0, 1, 2, 0, 3, -1 },
               { inf, -1, -1, 3, 2, 0, 0 }, { inf, 4, 3, inf, inf, 2, 0 } };

       int distance[][] = new int[noOfvertices][];
       for (int source = 0; source < noOfvertices; source++) {
           distance[source] = BellmanFord.bellmanFordEvaluation(source, adjacencymatrix);
       }

       for (int i = 0; i < noOfvertices; i++) {
           System.out.println(\"Displaying distance from source \'\" + i + \"\' to all the other vertices\");
           for (int j = 0; j < noOfvertices; j++) {
               System.out.print((distance[i][j] == inf ? \"inf\" : distance[i][j]) + \" \");
           }
           System.out.println(\"\ \");
       }
   }
}