Simple and Faster algorithm for Reachability in a Decremental Directed Graph

TL;DR

This paper introduces a simplified, faster randomized algorithm for maintaining source-sink reachability, single source reachability, and strongly connected components in decremental directed graphs, improving upon previous methods across various graph densities.

Contribution

The authors present a unified, simpler algorithm that outperforms prior algorithms for all graph densities in decremental reachability problems.

Findings

Achieves $ ilde{O}(mn^{4/5})$ total update time for $st$-Reachability.

Attains $ ilde{O}(n^{8/3})$ total update time for $st$-Reachability in dense graphs.

Provides $ ilde{O}(m n^{9/10})$ total update time for SSR and SCC.

Abstract

Consider the problem of maintaining source sink reachability($st$-Reachability), single source reachability(SSR) and strongly connected component(SCC) in an edge decremental directed graph. In particular, we design a randomized algorithm that maintains with high probability: 1) $st$-Reachability in $\tilde{O}(mn^{4/5})$ total update time. 2) $st$-Reachability in a total update time of $\tilde{O}(n^{8/3})$ in a dense graph. 3) SSR in a total update time of $\tilde{O}(m n^{9/10})$. 4) SCC in a total update time of $\tilde{O}(m n^{9/10})$. For all the above problems, we improve upon the previous best algorithm (by Henzinger et. al. (STOC 2014)). Our main focus is maintaining $st$-Reachability in an edge decremental directed graph (other problems can be reduced to $st$-Reachability). The classical algorithm of Even and Shiloach (JACM 81) solved this problem in $O(1)$ query time and $O(mn)$ total update time. Recently, Henzinger, Krinninger and Nanongkai (STOC 2014) designed a randomized algorithm which achieves an update time of $\tilde{O}(m n^{0.98})$ and broke the long-standing $O(mn)$ bound of Even and Shiloach. However, they designed four algorithms $A_i (1\le i \le 4)$ such that for graphs having total number of edges between $m_i$ and $m_{i+1}$ ($m_{i+1} > m_i$), $A_i$ outperforms other three algorithms. That is, one of the four algorithms may be faster for a particular density range of edges, but it may be too slow asymptotically for the other ranges. Our main contribution is that we design a {\it single} algorithm which works for all types of graphs. Not only is our algorithm faster, it is much simpler than the algorithm designed by Henzinger et.al. (STOC 2014).