Reliable Hubs for Partially-Dynamic All-Pairs Shortest Paths in Directed Graphs
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We give new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in O(mn^4/3W/ϵ) total time (where the edge weights are from [1,W]) and explicitly maintains a (1+ϵ)-approximate distance matrix. For a fixed ϵ>0, this is the first deterministic partially dynamic algorithm for all-pairs shortest paths in directed graphs, whose update time is o(n^2) regardless of the number of edges. Furthermore, we also show how to improve the state-of-the-art partially dynamic randomized algorithms for all-pairs shortest paths [Baswana et al. STOC'02, Bernstein STOC'13] from Monte Carlo randomized to Las Vegas randomized without increasing the running time bounds (with respect to the O(·) notation). Our results are obtained by giving new algorithms for the problem of dynamically maintaining hubs, that is a set of O(n/d) vertices which hit a shortest path between each pair of vertices, provided it has hop-length Ω(d). We give new subquadratic deterministic and Las Vegas algorithms for maintenance of hubs under either edge insertions or deletions.
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