You are given a black box Dijkstras implementation You are g

You are given a black box Dijkstra’s implementation

You are given a black box Dijkstra\'s implementation, Dscr, that takes as input a directed graph G = (V, E), vertices u, v epsilon V, and weighing function W: E rightarrow R^+ and computes the shortest path between the vertices u, v using the costs returned by W. It is common to want to compute the shortest path of a graph where the path-cost is the product of all the edge-costs in the path instead of the sum of the edge costs. However, since we already have an implementation of Dijkstra\'s using sum costs, we do not wish to duplicate work. Give an algorithm Pscr that on input Pscr(G, W, u, v) with W: E rightarrow Zopf^+, computes the shortest product path using Dscr to do most of the work.

Solution

/ A C / C++ program for Dijkstra\'s single source shortest path algorithm.

// The program is for adjacency matrix representation of the graph

  

#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 Source\ \");

   for (int i = 0; i < V; i++)

      printf(\"%d \\t\\t %d\ \", 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;

}