Improved Algorithms for Decremental Single-Source Reachability on Directed Graphs

Monika Henzinger, Sebastian Krinninger, Danupon Nanongkai

Abstract: Recently we presented the first algorithm for maintaining the set of nodes reachable from a source node in a directed graph that is modified by edge deletions with $$o(mn)$$ total update time, where $$m$$ is the number of edges and $$n$$ is the number of nodes in the graph [Henzinger et al. STOC 2014]. The algorithm is a combination of several different algorithms, each for a different $$m$$ vs.  $$n$$ trade-off. For the case of $$m = \varTheta (n^{1.5})$$ the running time is $$O(n^{2.47})$$ , just barely below $$mn = \varTheta (n^{2.5})$$ . In this paper we simplify the previous algorithm using new algorithmic ideas and achieve an improved running time of $$\tilde{O}(\min ( m^{7/6} n^{2/3}, m^{3/4} n^{5/4 + o(1)}, m^{2/3} n^{4/3+o(1)} + m^{3/7} n^{12/7+o(1)}))$$ . This gives, e.g., $$O(n^{2.36})$$ for the notorious case $$m = \varTheta (n^{1.5})$$ . We obtain the same upper bounds for the problem of maintaining the strongly connected components of a directed graph undergoing edge deletions. Our algorithms are correct with high probabililty against an oblivious adversary.