Improved Algorithms for Decremental Single-Source Reachability on Directed Graphs

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 $m=\Theta (n^{1.5})$ the running time is $O(n^{2.47})$, just barely below $mn=\Theta (n^{2.5})$. 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 $m=\Theta (n^{1.5})$. 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.

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:1504.07959