Improved Algorithms for Decremental Single-Source Reachability on Directed Graphs

TL;DR

This paper presents simplified and improved algorithms for decremental single-source reachability and strongly connected components in directed graphs, achieving faster update times with high probability against oblivious adversaries.

Contribution

The authors simplify previous algorithms and improve the running time bounds for decremental reachability and SCC maintenance in directed graphs.

Findings

Achieved an update time of O( m^{7/6} n^{2/3}) for decremental reachability.

Provided bounds of O( m^{3/4} n^{5/4 + o(1)}) and O( m^{2/3} n^{4/3+o(1)} + m^{3/7} n^{12/7+o(1)}) for different graph sizes.

Algorithms are correct with high probability against an oblivious adversary.

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.