Question

Design and implement Dijkstra’s algorithm to compute all-pair shortest paths in any given graph using

An adjacency matrix using a one-dimensional array for storing only the elements of the lower triangle in the adjacency matrix.[Program in C language]

The input to program must be connected, undirected, and weighted graphs. The programs must be able to find shortest paths on two types of connected, undirected, and weighted graphs: complete graph (a graph with a link between every pair of nodes) and sparse graph (a graph with exactly n-1 links). To prepare graphs as inputs must include these additional programs (functions) as follows:

1. Random. A program to randomly generate a number between 1 and n. You can use a random number generator function in the libraries of your programming language or write your own function.

2. Complete. A program that can create an adjacency matrix for an undirected complete graph with any given size n.

3. Sparse. A program that can create an adjacency matrix for an undirected sparse graph with any given size n. The Sparse program must use the Random program (number 1 in this list) to determine all the links (n-1) in the sparse graph.

4. Weight. To include weights on the links of the undirected complete graphs (generated by the Complete program, number 2 in this list) and on the links of the undirected sparse graphs (generated by the Sparse program, number 3 in this list), use the Random program (number 1 in this list) to generate positive numbers as weights

[Need outputs and graphs also in the solution along with source code]

TEST CASE:

Test case:

adj-test case-1

. . . 29 . . . .

. . . . . 11 11 .

. . . 12 . 5 5 .

29 . 12 . 5 . 13 .

. . . 5 . . 7 11

. 11 5 . . . . 17

. 11 5 13 7 . . .

. . . . 11 17 . .

adj-test case-2

. . . 29 . . . .

. . . . . 11 11 .

. . . 12 . 5 5 .

29 . 12 . 5 . 13 .

. . . 5 . . 7 11

. 11 5 . . . . 17

. 11 5 13 7 . . .

. . . . 11 17 . .

10 15 Shortest path between v2 - v8: The distance is: 35 The path is: v2->v3->v6->v9->v8 13 25 25 17 28 25 14 18 Shortest path between v12-v10: The distance is:? The path is:? 14 12 29 13 16 12

Answer 1

 
#include <stdio.h>
#include <limits.h>
 
// Number of vertices in the graph
#define V 9
 
// A utility function to find the vertex with minimum distance value, from
// the set of vertices not yet included in shortest path tree
int minDistance(int dist[], bool sptSet[])
{
   // Initialize min value
   int min = INT_MAX, min_index;
 
   for (int v = 0; v < V; v++)
     if (sptSet[v] == false && dist[v] <= min)
         min = dist[v], min_index = v;
 
   return min_index;
}
 
// A utility function to print the constructed distance array
int printSolution(int dist[], int n)
{
   printf("Vertex   Distance from Sourcen");
   for (int i = 0; i < V; i++)
      printf("%d tt %dn", i, dist[i]);
}
 
// Funtion that implements Dijkstra's single source shortest path algorithm
// for a graph represented using adjacency matrix representation
void dijkstra(int graph[V][V], int src)
{
     int dist[V];     // The output array.  dist[i] will hold the shortest
                      // distance from src to i
 
     bool sptSet[V]; // sptSet[i] will true if vertex i is included in shortest
                     // path tree or shortest distance from src to i is finalized
 
     // Initialize all distances as INFINITE and stpSet[] as false
     for (int i = 0; i < V; i++)
        dist[i] = INT_MAX, sptSet[i] = false;
 
     // Distance of source vertex from itself is always 0
     dist[src] = 0;
 
     // Find shortest path for all vertices
     for (int count = 0; count < V-1; count++)
     {
       // Pick the minimum distance vertex from the set of vertices not
       // yet processed. u is always equal to src in first iteration.
       int u = minDistance(dist, sptSet);
 
       // Mark the picked vertex as processed
       sptSet[u] = true;
 
       // Update dist value of the adjacent vertices of the picked vertex.
       for (int v = 0; v < V; v++)
 
         // Update dist[v] only if is not in sptSet, there is an edge from 
         // u to v, and total weight of path from src to  v through u is 
         // smaller than current value of dist[v]
         if (!sptSet[v] && graph[u][v] && dist[u] != INT_MAX 
                                       && dist[u]+graph[u][v] < dist[v])
            dist[v] = dist[u] + graph[u][v];
     }
 
     // print the constructed distance array
     printSolution(dist, V);
}
 
// driver program to test above function
int main()
{
   /* Let us create the example graph discussed above */
   int graph[V][V] = {{0, 4, 0, 0, 0, 0, 0, 8, 0},
                      {4, 0, 8, 0, 0, 0, 0, 11, 0},
                      {0, 8, 0, 7, 0, 4, 0, 0, 2},
                      {0, 0, 7, 0, 9, 14, 0, 0, 0},
                      {0, 0, 0, 9, 0, 10, 0, 0, 0},
                      {0, 0, 4, 14, 10, 0, 2, 0, 0},
                      {0, 0, 0, 0, 0, 2, 0, 1, 6},
                      {8, 11, 0, 0, 0, 0, 1, 0, 7},
                      {0, 0, 2, 0, 0, 0, 6, 7, 0}
                     };
 
    dijkstra(graph, 0);
 
    return 0;
}