Decremental transitive closure and shortest paths for planar digraphs and beyond

Abstract

In this paper we show that the tools used to obtain the best state-of-the-art decremental algorithms for reachability and approximate shortest paths in directed graphs can be successfully combined with the existence of small separators in certain graph classes. In particular, for graph classes admitting balanced separators of size O( p n), such as planar, bounded-genus and minor-free graphs, we show that for both transitive closure and (1 + ϵ)-approximate all pairs shortest paths (where ϵ is constant), there exist decremental algorithms with Õ(n3/2) total update time and Õ( p n) worst-case query time. Additionally, for the case of planar graphs, we show that for any t 2 [1; n], there exists a decremental transitive closure algorithm with Õ(n2/t) total update time and Õ( p t) worst-case query time. In particular, for t/n2/3, if all the edges are eventually deleted, we obtain Õ(n1/3) amortized update and query times. Most of the algorithms we obtain are correct with high probability against an oblivious adversary.