Simulating Floyd's Algorithm in MATLAB

In this text, we will implement Floyd's algorithm simulation using MATLAB to calculate the shortest paths and routing information between any two nodes in a graph. Floyd's algorithm is a highly practical graph algorithm that finds the shortest paths between all pairs of nodes in either directed or undirected graphs. The algorithm employs dynamic programming principles and utilizes three nested loops with matrix operations to store inter-node distances and path information. For the MATLAB implementation, we will leverage its powerful matrix computation capabilities to execute the algorithm efficiently. The core implementation involves initializing a distance matrix representing edge weights (with infinite values for non-connected nodes) and a path matrix to track intermediate nodes. The algorithm then iteratively updates these matrices using the key recurrence relation: D[i,j] = min(D[i,j], D[i,k] + D[k,j]) for each intermediate node k. We will enhance the simulation with visualization components to display the graph structure and resulting shortest paths, along with data output functions to show the computed distance matrix and routing information. Additionally, we will analyze Floyd's algorithm advantages (O(n³) time complexity for dense graphs, simplicity of implementation) and limitations (computational intensity for large graphs), and discuss potential applications to related problems such as network routing optimization and transportation system planning. The implementation will include error handling for invalid inputs and validation checks for graph connectivity properties.