Question

i need to find complexity and cost and runtime for each line of the following c++ code :

// A C++ program for Dijkstra's single source shortest path algorithm.
// The program is for adjacency matrix representation of the graph

#include <limits.h>
#include <stdio.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
void printSolution(int dist[])
{
   printf("Vertex \t\t Distance from Source\n");
   for (int i = 0; i < V; i++)
       printf("%d \t\t %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);
}

Answer 1

#include <limits.h>
#include <stdio.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
//complexity is O(V)
//max of V elements accessed , so o(v) as v operation will b performed so o(v)
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
// complexity is o(V) as each element is accessed only once and total of V elements are accessed or its O(V)
void printSolution(int dist[])
{
printf("Vertex \t\t Distance from Source\n");
for (int i = 0; i < V; i++)
printf("%d \t\t %d\n", i, dist[i]);
}

// Function that implements Dijkstra's single source shortest path algorithm
// for a graph represented using adjacency matrix representation
//complexity will be O(V) *[ O(V) + O(V) ] = O(V) /(as O(V) + O(V) = O(V))
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; // complexity to initailize each element is O(1) , so we need to initalize V element its will be o(V)

// 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++) { //total of V element accessed so o(v)
// 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); //complexity to calculate minDistance is o(V) as discussed above.

// 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++) //total of V elements accessed so its o(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);
}