Implementation of Prim's Algorithm and Kruskal's Algorithm in MATLAB

In this documentation, we will discuss how to implement Prim's algorithm and Kruskal's algorithm using MATLAB. The implementation requires two input parameters: x representing the adjacency matrix and p representing the node set. The output will be the spanning tree edge set G. Both algorithms are used for computing minimum spanning trees, which have significant applications in graph theory. For Prim's algorithm implementation, we initialize with a single vertex and iteratively add the minimum-weight edge connecting the growing tree to vertices not yet included. This greedy approach requires maintaining a priority queue to efficiently select the next edge. Kruskal's algorithm implementation involves sorting all edges by weight and adding them to the spanning tree in ascending order, while ensuring no cycles are formed using a union-find data structure for efficient cycle detection. We will provide detailed explanations of the implementation steps, including key MATLAB functions such as graph initialization, edge sorting, and cycle checking mechanisms. The discussion will cover algorithmic complexities, advantages and disadvantages of each approach, and their appropriate application scenarios. Through this documentation, you will gain deeper understanding of these algorithms and be able to apply them effectively in your projects and studies. The code implementation will include proper error handling for invalid inputs and visualization options for the resulting spanning trees.