Decremental Strongly-Connected Components and Single-Source Reachability in Near-Linear Time

Abstract: Computing the Strongly-Connected Components (SCCs) in a graph G=(V,E) is known to take only O(m + n) time using an algorithm by Tarjan from 1972[SICOMP 72] where m = |E|, n=|V|. For fully-dynamic graphs, conditional lower bounds provide evidence that the update time cannot be improved by polynomial factors over recomputing the SCCs from scratch after every update. Nevertheless, substantial progress has been made to find algorithms with fast update time for *decremental* graphs, i.e. graphs that undergo edge deletions.

In this paper, we present the first algorithm for general decremental graphs that maintains the SCCs in total update time O(m polylog(n)), thus only a polylogarithmic factor from the optimal running time. Previously such a result was only known for the special case of planar graphs [Italiano et al, STOC 17]. Our result should be compared to the formerly best algorithm for general graphs achieving O(m\sqrt{n} polylog(n)) total update time by Chechik et.al. [FOCS 16] which improved upon a breakthrough result leading to O(mn^{0.9 + o(1)}) total update time by Henzinger, Krinninger and Nanongkai [STOC 14, ICALP 15]; these results in turn improved upon the longstanding bound of O(mn) by Roditty and Zwick [STOC 04].