Incremental Algorithms for the Single-Source Shortest Path Problem

Daniele Frigioni, Alberto Marchetti-Spaccamela, Umberto Nanni

Abstract: We consider the problem of updating a single source shortest path tree in either a directed or an undirected graph, with positive real edge weights. Our semidynamic algorithms for the incremental problem (handling edge insertions and cost decrements) work for any graph, but their performances depend on the class of the considered graph. In any case our algorithms have optimal space requirements and query time. The cost of updates is computed in terms of amortized output complexity. In the case of graphs with linear size genus (including planar graphs), bounded degree graphs and bounded treewidth graphs, the incremental algorithm requires O(logn) amortized time per vertex update, where a vertex is considered updated if reduces its distance from the source. We also consider the decremental problem for planar graphs, providing algorithms and data structures with analogous performances. We remark that our algorithms, based on Di-jkstra's techniques [5], require simple data structures that are really suitable for a practical and straightforward implementation.