Dijkstra's Algorithm

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.

REFERENCES

Dijkstra, E. W. "A Note on Two Problems in Connection with Graphs." Numerische Math. 1, 269-271, 1959.

Skiena, S. "Dijkstra's Algorithm." §6.1.1 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 225-227, 1990.

Whiting, P. D. and Hillier, J. A. "A Method for Finding the Shortest Route through a Road Network." Operational Res. Quart. 11, 37-40, 1960.