Efficient Algorithms for Shortest Paths in Sparse Networks

Algorithms for finding shortest paths are presented which are faster than algorithms previously known on networks which are relatively sparse in arcs. Known results which the results of this paper extend are surveyed briefly and analyzed. A new implementation for priority queues is employed, and a class of “arc set partition” algorithms is introduced. For the single source problem on networks with nonnegative arcs a running time of <italic>O</italic>(min(<italic>n</italic><supscrpt>1+1/<italic>k</italic></supscrpt> + <italic>e</italic>, <italic>n</italic> + <italic>e</italic>) log <italic>n</italic>)) is achieved, where there are <italic>n</italic> nodes and <italic>e</italic> arcs, and <italic>k</italic> is a fixed integer satisfying <italic>k</italic> > 0. This bound is <italic>O</italic>(<italic>e</italic>) on dense networks. For the single source and all pairs problem on unrestricted networks the running time is <italic>O</italic>(min(<italic>n</italic><supscrpt>2+1/<italic>k</italic></supscrpt> + <italic>ne</italic>, <italic>n</italic><supscrpt>2</supscrpt> log <italic>n</italic> + <italic>ne</italic> log <italic>n</italic>).

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