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=Θ(n1.5) the
running time is O(n2.47), just barely below mn=Θ(n2.5). In
this paper we simplify the previous algorithm using new algorithmic ideas and
achieve an improved running time of O~(min(m7/6n2/3,m3/4n5/4+o(1),m2/3n4/3+o(1)+m3/7n12/7+o(1))). This gives,
e.g., O(n2.36) for the notorious case m=Θ(n1.5). 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.Comment: This paper was presented at the International Colloquium on Automata,
Languages and Programming (ICALP) 2015. A full version combining the findings
of this paper and its predecessor [Henzinger et al. STOC 2014] is available
at arXiv:1504.0795