Finding Minimum Paths Using Dijkstra's Algorithm

Dijkstra's algorithm is a fundamental graph theory algorithm used to find the shortest path between a start node and a target node in weighted graphs. The algorithm's core methodology involves iteratively expanding the set of known shortest paths to achieve a globally optimal solution through a greedy approach.

The algorithm initializes by setting the start node's distance to 0 and all other nodes' distances to infinity. In each iteration, it selects the unprocessed node with the minimum current distance, then updates the shortest distances of its neighboring nodes. When a better path is discovered during updates, the algorithm records new distances and predecessor nodes. This process continues until all reachable nodes are processed, typically implemented using a priority queue for efficient minimum-distance node selection.

An adjacency matrix serves as one method for representing graph structures, where rows and columns correspond to nodes, and matrix values indicate edge weights between nodes. In Dijkstra's implementation, the adjacency matrix enables efficient lookup of connections and weights between any two nodes, though adjacency lists may be preferred for sparse graphs to optimize space complexity.

The algorithm not only computes the shortest distance from start to target but also reconstructs the exact path by backtracking through predecessor nodes. This dual capability makes Dijkstra's algorithm valuable both for theoretical studies and practical applications like network routing protocols and transportation navigation systems, where path reconstruction is implemented through parent pointer arrays or node tracing functions.