Dijkstra's algorithm is an algorithm for finding a graph geodesic, i.e., the shortest path between two graph vertices in a graph. It functions by constructing a shortest-path tree from the initial vertex to every other vertex in the graph.
The algorithm is implemented in the Wolfram Language as FindShortestPath[g, Method -> "Dijkstra"].
The worst-case running time for the Dijkstra algorithm on a graph with 
nodes and 
edges is 
because it allows for directed cycles. It even finds
the shortest paths from a source node 
to all other nodes in the graph. This is basically 
for node selection and 
for distance updates. While 
is the best possible complexity for dense graphs, the
complexity can be improved significantly for sparse graphs.
With slight modifications, Dijkstra's algorithm can be used as a reverse algorithm that maintains minimum spanning trees for the sink node. With further modifications, it can be extended to become bidirectional.
The bottleneck in Dijkstra's algorithm is node selection. However, using Dial's implementation, this can be significantly improved for sparse graphs.
In the Season 3 episode "Money For Nothing" (2007) of the television crime drama NUMB3RS, mathematics professor Charlie Eppes uses Dijkstra's algorithm to find the most likely escape routes out of Los Angeles for a hijacked truck that is carrying millions of dollars in cash and medical supplies and also two kidnapping victims.
Haeupler et al. (2024) proved that Dijkstra's algorithm is universally optimal both in its running time and number of comparisons when combined with a sufficiently efficient heap data structure. This result provides a powerful beyond-worst-case performance guarantee for graph algorithms. Informally speaking, it means that Dijkstra's algorithm performs as well as possible for every possible graph topology (Haeupler et al. 2024).
