Kruskal Algorithm

 

What Is Kruskal’s Algorithm?

  • Kruskal’s Algorithm is a greedy algorithm used to find the Minimum Spanning Tree (MST) of a connected, undirected, weighted graph.

  • It builds the MST by selecting the smallest-weight edges first, ensuring no cycles are formed.

 Why Use Kruskal’s Algorithm?

  • It guarantees the least total edge weight to connect all vertices.

  • Especially useful when the graph has sparse connections or edges are already sorted.

  • It’s simple and efficient for edge-centric problems.

 Where to Use Kruskal’s Algorithm?

  • Network Design: Laying out cables or pipelines with minimal cost.

  • Road Systems: Connecting cities with shortest total distance.

  • Clustering Algorithms: In machine learning, for hierarchical clustering.

  • Electrical Grids: Designing efficient power distribution networks.

 Key Steps in Kruskal’s Algorithm

  1. Sort all edges by weight (ascending).

  2. Initialize MST as an empty set.

  3. Iterate through edges:

    • Add edge if it doesn’t form a cycle (using Union-Find).

  4. Stop when MST has V-1 edges (where V = number of vertices).

 Constraints

  • Graph must be connected, undirected, and weighted.

  • Cannot handle negative cycles, but can handle negative weights.

  • Requires Union-Find (Disjoint Set) for cycle detection.

 Time Complexity

  • O(E log E) → Sorting edges dominates, where E = number of edges.

  • Union-Find operations are nearly constant time with path compression.

 Space Complexity

  • O(V + E) → For storing edges and disjoint sets, where V = vertices.

These complexities hold across best, average, and worst cases due to the sorting step and disjoint-set operations.

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