All-Pair Shortest Path. You’ve been promoted to the head of logistics for the Ga

Question

All-Pair Shortest Path. You’ve been promoted to the head of logistics for the Galactic Empire. The Rebel Alliance has disrupted the troop-transport routes between your outposts on various planets, so the first order of business is find new ones. Given the galactic map of possible routes below (distances are in light-years), use your favorite all-pairs, shortest paths algorithm to calculate the best routes between each of the outposts. Be sure show your work.

Ariel Minbar 2.4K 1.5K Hoth 3.5K 3.3K. o'noS Narn Romulus 4.0K 2.3K 1.2K 3.0K 6.4K 4.4K 6.0K 4.5K Dagabah 4.0K Vulcan 2.4K Gallifrey New Santo Earth

Explanation / Answer

First we label vertices names like-

0-Narn

1-Arial

2-Hoth

3-Minbar

4-Qo'nos

5-Gallifray

6-Santo

7-Vulcan

8-New Earth

9-Romulus

10-DagaBah

Second we create AdjacencyMatrix of the above graph(I assume you know how to create AdjacencyMatrix).

Since there are 11 vertex our matrix will be of 11*11

The solution in C++ will be-

// 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 11

// 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(float dist[], bool sptSet[])
{
// Initialize min value
float min = INT_MAX;
int 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(float dist[], int n)
{
printf("Vertex Distance from Source ");
for (int i = 0; i < V; i++)
   printf("%d %d ", i, dist[i]);
}

// Funtion that implements Dijkstra's single source shortest path algorithm
// for a graph represented using adjacency matrix representation
void dijkstra(float graph[V][V], int src)
{
   float 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 */
float graph[V][V] = {
{0, 3.3, 0, 0, 0, 0, 0,0,0,0, 2.3},
{3.3, 0, 1.4, 0, 0, 0, 0, 4, 6, 0, 0},
{0, 1.5, 0, 3.5, 0,0,0,3, 0, 0, 0},
{0, 0, 3.5, 0, 2.4, 6.4, 4.5, 0, 0 , 0, 0},
{0, 0, 0, 2.4, 0, 4.4, 0, 0, 0,0,0},
{0, 0, 0, 6.4, 4.4, 0, 0,0 ,0, 0, 0},
{0, 0, 0, 4.5, 0, 0, 0, 2.4,0,0,0},
{0, 4, 3, 0, 0, 0, 2.4, 0, 0,0,0},
{0, 8, 0, 0, 0, 0, 0, 0, 0,4,0},
{0,0,0,0,0,0,0,0,4,0,1.2},
{2.3,0,0,0,0,0,0,0,0,1.2,0}
};
  
   dijkstra(graph, 0);

   return 0;
}

Due to limited time unit conversion has been ignored and also ou might get wrong format error.

0 3.3 0 0 0 0 0 0 0 0 2.3 3.3 0 1.5 0 0 0 0 4 6 0 0 0 1.5 0 3.5 0 0 0 3 0 0 0 0 0 3.5 0 2.4 6.4 4.5 0 0 0 0 0 0 0 2.4 0 4.4 0 0 0 0 0 0 0 0 6.4 4.4 0 0 0 0 0 0 0 0 0 4.5 0 0 0 2.4 0 0 0 0 4 3 0 0 0 2.4 0 0 0 0 0 8 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 1.2 2.3 0 0 0 0 0 0 0 0 1.2 0

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