MATLAB Implementation of Dijkstra's Algorithm for Shortest Path Finding

Dijkstra's algorithm is a classical algorithm for finding single-source shortest paths in graphs, proposed by Dutch computer scientist Edsger Dijkstra in 1956. This algorithm works on weighted directed or undirected graphs and requires that edge weights be non-negative.

Implementing Dijkstra's algorithm in MATLAB can follow this approach:

Initialization Define the graph's adjacency matrix where each matrix element represents the weight between nodes. If no direct connection exists between two nodes, set the value to infinity (`Inf`). Create two arrays: one to record the current shortest distance from the start node to each node (initially set to infinity), and another to track whether each node has been visited (initially unvisited). Set the distance from the start node to itself as 0.

Priority Queue Processing Use a priority queue (or manual iteration) to select the unvisited node with the smallest current distance as the current node. Mark this node as visited.

Update Neighboring Node Distances Iterate through all neighbors of the current node. Calculate the distance from the start node to each neighbor through the current node. If this distance is shorter than the previously recorded distance, update the shortest distance array.

Loop Until All Nodes Are Visited Repeat the above process until all nodes have been visited, ultimately obtaining the shortest paths from the start node to all other nodes.

Optimization and Extensions The node selection step can be optimized using a priority queue (such as a min-heap) to improve algorithm efficiency. For sparse graphs, adjacency lists may be more memory-efficient than adjacency matrices. This algorithm can be applied to practical scenarios like path planning and network routing.

The time complexity of Dijkstra's algorithm depends on the implementation. With an adjacency matrix, it's typically O(n²), while using a priority queue can optimize it to O(n log n + m), where n is the number of nodes and m is the number of edges. In MATLAB, leveraging matrix operations effectively can further enhance performance.