Implementation of Dijkstra's Algorithm for Shortest Path Problems

Dijkstra's algorithm is a classic solution for shortest path problems, applicable to both directed and undirected graphs with non-negative edge weights. The algorithm's core methodology involves iteratively expanding the set of vertices with known shortest paths from the source vertex until all vertices are processed.

The implementation follows this logical flow: First, initialize a distance array to track the current shortest distances from the source vertex. The source vertex's distance is set to 0, while all other vertices are initialized with infinity (or a sufficiently large value). The algorithm then repeatedly selects the unprocessed vertex with the minimum current distance, and updates the distances of its adjacent vertices through relaxation - a process where we compare the current known distance with the potential new path through the selected vertex. This iterative process continues until all vertices have been processed.

In practical implementations, Dijkstra's algorithm typically utilizes a priority queue (such as a min-heap) to efficiently extract the vertex with the smallest distance. The key operations involve maintaining the priority queue with O(log n) complexity for both extraction and distance updates. Crucial implementation details include: maintaining a visited set to track processed vertices, handling graph representations (adjacency list/matrix), and implementing the relaxation condition: if distance[u] + weight(u,v) < distance[v], then update distance[v]. The algorithm requires non-negative edge weights to guarantee correctness, as negative weights could create cycles that continuously reduce path distances.

Dijkstra's algorithm has extensive applications in network routing protocols, transportation navigation systems, and network optimization problems. It serves as one of the most fundamental and important algorithms in graph theory, with time complexity of O((V+E) log V) when properly implemented with a priority queue.