Kruskal's Algorithm for Minimum Spanning Tree: Implementation and Analysis

Kruskal's algorithm is a classic greedy approach for constructing minimum spanning trees, which operates by progressively selecting edges with the smallest weights that do not form cycles until all vertices are connected. MATLAB serves as an ideal computational environment for implementing this algorithm efficiently due to its robust matrix operations and built-in functions.

The algorithm workflow consists of three key phases: First, sort all edges in ascending order of their weights using MATLAB's sort functions (e.g., sortrows). Second, initialize a union-find (disjoint set) data structure to detect cycles. Third, iterate through the sorted edges—if endpoints belong to different sets, merge them using union operations and add the edge to the spanning tree. In MATLAB, arrays can simulate union-find operations efficiently, where find-set operations determine connectivity and union operations merge components.

Critical implementation considerations include: applying path compression in union-find to optimize time complexity, and verifying graph connectivity to prevent premature termination. The algorithm extends to weighted undirected graphs and network optimization scenarios. Time complexity is dominated by the sorting step, typically O(E log E), where E represents the number of edges.