Prim's Algorithm

Write a C++ program to implement Prim's algorithm to find the minimum spanning tree (MST) of a connected, undirected graph with weighted edges. The program should output the edges of the MST along with their weights.

Example 1

Input: 
   Number of vertices: 4
   Number of edges: 5
   Edges and weights:
   0 1 10
   0 2 6
   0 3 5
   1 3 15
   2 3 4
Output: 
   Edges in the MST:
   0 - 3 : 5
   3 - 2 : 4
   0 - 1 : 10
Explanation:
   

   
Step 1: Initialize MST as empty and a priority queue to store vertices.
   
Step 2: Start with vertex 0 (can start with any vertex).
   
Step 3: Add all edges from vertex 0 to the priority queue (edge weights: 10, 6, 5).
   
Step 4: Choose the smallest edge (0-3) with weight 5 and add it to MST.
   
Step 5: Add all edges from vertex 3 to the priority queue (edge weights: 15, 4).
   
Step 6: Choose the smallest edge (3-2) with weight 4 and add it to MST.
   
Step 7: Choose the smallest remaining edge (0-1) with weight 10 and add it to MST.
   
Step 8: Return the MST with total weight 5 + 4 + 10 = 19.
   

Example 2

Input: 
   Number of vertices: 5
   Number of edges: 7
   Edges and weights:
   0 1 2
   0 3 6
   1 2 3
   1 3 8
   1 4 5
   2 4 7
   3 4 9
Output: 
   Edges in the MST:
   0 - 1 : 2
   1 - 2 : 3
   1 - 4 : 5
   0 - 3 : 6
Explanation:
   

   
Step 1: Initialize MST as empty and a priority queue to store vertices.
   
Step 2: Start with vertex 0 (can start with any vertex).
   
Step 3: Add all edges from vertex 0 to the priority queue (edge weights: 2, 6).
   
Step 4: Choose the smallest edge (0-1) with weight 2 and add it to MST.
   
Step 5: Add all edges from vertex 1 to the priority queue (edge weights: 3, 8, 5).
   
Step 6: Choose the smallest edge (1-2) with weight 3 and add it to MST.
   
Step 7: Choose the smallest edge (1-4) with weight 5 and add it to MST.
   
Step 8: Choose the smallest edge (0-3) with weight 6 and add it to MST.
   
Step 9: Return the MST with total weight 2 + 3 + 5 + 6 = 16.
   

Constraints

1 ≤ Number of vertices ≤ 100
1 ≤ Number of edges ≤ 1000
Time Complexity: O(V^2) or O(E log V)
Space Complexity: O(V + E)