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Kruskal's Algorithm

Certification: Advanced Level Accuracy: 0% Submissions: 0 Points: 15

Write a C++ program to implement Kruskal'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:
   2 - 3 : 4
   0 - 3 : 5
   0 - 1 : 10
    • 

  • Explanation:
   
      
   
    • Step 1: Sort all edges in non-decreasing order of their weight.
      • 
   
    • Step 2: Initialize an empty MST and a disjoint set data structure.
      • 
   
    • Step 3: Pick edges in increasing order of weight:
      
        
      
      • Edge (2-3) with weight 4: Include in MST
        • 
      
      • Edge (0-3) with weight 5: Include in MST
        • 
      
      • Edge (0-2) with weight 6: Skip (would create a cycle)
        • 
      
      • Edge (0-1) with weight 10: Include in MST
        • 
      
      • Edge (1-3) with weight 15: Skip (would create a cycle)
        • 
      
      
   
      • 
   
    • Step 4: Return the edges included in the MST with total weight 4 + 5 + 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: Sort all edges in non-decreasing order of their weight.
      • 
   
    • Step 2: Initialize an empty MST and a disjoint set data structure.
      • 
   
    • Step 3: Pick edges in increasing order of weight:
      
        
      
      • Edge (0-1) with weight 2: Include in MST
        • 
      
      • Edge (1-2) with weight 3: Include in MST
        • 
      
      • Edge (1-4) with weight 5: Include in MST
        • 
      
      • Edge (0-3) with weight 6: Include in MST
        • 
      
      • Edge (2-4) with weight 7: Skip (would create a cycle)
        • 
      
      • Edge (1-3) with weight 8: Skip (would create a cycle)
        • 
      
      • Edge (3-4) with weight 9: Skip (would create a cycle)
        • 
      
      
   
      • 
   
    • Step 4: Return the edges included in the MST with total weight 2 + 3 + 5 + 6 = 16.
      • 
   
    

    •
Constraints
    

  • 1 ≤ Number of vertices ≤ 100
    • 

  • 1 ≤ Number of edges ≤ 1000
    • 

  • Time Complexity: O(E log E) or O(E log V)
    • 

  • Space Complexity: O(V + E)
    •
SetTreePwCSamsung
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Hint
Solution

Solution Hints

     
  • Use a disjoint-set (Union-Find) data structure to manage the vertices and detect cycles.
    •  
  • Sort all the edges in non-decreasing order of their weight.
    •  
  • Iterate through the sorted edges and add them to the MST if they do not form a cycle.
    •  
  • Continue until there are (V-1) edges in the MST.
    •

Steps to solve by this approach:

 Step 1: Sort all edges in non-decreasing order of their weight.

 Step 2: Initialize result set as empty and create disjoint sets for each vertex.

 Step 3: For each edge in sorted order, check if including it creates a cycle.

 Step 4: Use union-find algorithm to detect if both endpoints are in same set (would create cycle).

 Step 5: If no cycle is formed, add the edge to result and merge sets of the two vertices.

 Step 6: Repeat until we have V-1 edges in the result (where V is number of vertices).

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