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Minimum Spanning Tree (MST)

Certification: Advanced Level Accuracy: 66.67% Submissions: 3 Points: 15

Write a Python function that implements Kruskal's algorithm to find a connected, undirected graph's Minimum Spanning Tree (MST). The MST is a subset of the edges that connects all vertices with the minimum possible total edge weight.

Example 1
    

  • Input: Edges = [(0, 1, 10), (0, 2, 6), (0, 3, 5), (1, 3, 15), (2, 3, 4)], Vertices = 4
    • 

  • Output: [(2, 3, 4), (0, 3, 5), (0, 2, 6)]
    • 

  • Explanation: 

      

    • Step 1: Sort all edges in non-decreasing order by weight: (2,3,4), (0,3,5), (0,2,6), (0,1,10), (1,3,15).
      • 

    • Step 2: Pick edges one by one and add to MST if they don't form a cycle.
      • 

    • Step 3: First edge (2,3,4) is added.
      • 

    • Step 4: Second edge (0,3,5) is added.
      • 

    • Step 5: Third edge (0,2,6) is added.
      • 

    • Step 6: No need to add more edges as all vertices are already connected.
      • 

    • Step 7: The MST consists of edges [(2,3,4), (0,3,5), (0,2,6)] with total weight 15.
      • 

    

    •
Example 2
    

  • Input: Edges = [(0, 1, 4), (0, 2, 3), (1, 2, 1), (1, 3, 2), (2, 3, 4)], Vertices = 4
    • 

  • Output: [(1, 2, 1), (1, 3, 2), (0, 2, 3)]
    • 

  • Explanation: 

      

    • Step 1: Sort all edges in non-decreasing order by weight: (1,2,1), (1,3,2), (0,2,3), (0,1,4), (2,3,4).
      • 

    • Step 2: Pick edges one by one and add to MST if they don't form a cycle.
      • 

    • Step 3: First edge (1,2,1) is added.
      • 

    • Step 4: Second edge (1,3,2) is added.
      • 

    • Step 5: Third edge (0,2,3) is added.
      • 

    • Step 6: No need to add more edges as all vertices are already connected.
      • 

    • Step 7: The MST consists of edges [(1,2,1), (1,3,2), (0,2,3)] with total weight 6.
      • 

    

    •
Constraints
    

  • 1 ≤ number of vertices ≤ 10^5
    • 

  • 0 ≤ number of edges ≤ 10^6
    • 

  • 1 ≤ edge weights ≤ 10^6
    • 

  • Graph is connected
    • 

  • Time Complexity: O(E log E), where E is the number of edges
    • 

  • Space Complexity: O(V + E), where V is the number of vertices and E is the number of edges
    •
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Hint
Solution

Solution Hints

    
  
  • Sort all edges in non-decreasing order of their weight
    • 
  
  • Use a disjoint set (Union-Find) data structure to detect cycles
    • 
  
  • Process edges in order of increasing weight
    • 
  
  • For each edge, if including it doesn't form a cycle, add it to the MST
    • 
  
  • Return the list of 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 a disjoint set data structure with Union-Find operations.

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

 Step 4: If no cycle is formed, include the edge in the MST and union the sets.

 Step 5: Continue until V-1 edges are included (where V is the number of vertices).
 Step 6: Return the edges of the minimum spanning tree.

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