3_dijkstra.cpp

/*
Below are the detailed steps used in Dijkstra’s algorithm to find 
the shortest path from a single source vertex to all other vertices in the given graph.
Algorithm
1) Create a set sptSet (shortest path tree set) that keeps track of vertices included
   in shortest path tree, i.e., whose minimum distance from source is calculated and finalized.
   Initially, this set is empty.
2) Assign a distance value to all vertices in the input graph. Initialize all distance values as INFINITE.
   Assign distance value as 0 for the source vertex so that it is picked first.
3) While sptSet doesn’t include all vertices
….a) Pick a vertex u which is not there in sptSet and has minimum distance value.
….b) Include u to sptSet.
….c) Update distance value of all adjacent vertices of u. 
     To update the distance values, iterate through all adjacent vertices.
	 For every adjacent vertex v, if sum of distance value of u (from source) 
	 and weight of edge u-v, is less than the distance value of v, then update the distance value of v.
*/

// 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\n"); 
   for (int i = 0; i < V; i++) 
      printf("%d tt %d\n", i, dist[i]); 
} 
   
// Function 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 be 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 the 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; 
}