Using Read-$k$ Inequalities to Analyze a Distributed MIS Algorithm

TL;DR

This paper extends the analysis of distributed MIS algorithms from trees to bounded arboricity graphs using read-$k$ inequalities, resulting in faster algorithms for certain graph classes.

Contribution

It introduces a new probabilistic analysis technique using read-$k$ inequalities to analyze MIS algorithms on bounded arboricity graphs.

Findings

The algorithm runs in $O( ext{poly}( ext{arboricity}) imes \u221a{ ext{log} n imes ext{log} ext{log} n})$ rounds.

For small arboricity, the algorithm outperforms previous bounded arboricity MIS algorithms.

The analysis leverages a tail inequality for read-$k$ families of random variables.

Abstract

Until recently, the fastest distributed MIS algorithm, even for simple graphs, e.g., unoriented trees has been the simple randomized algorithm discovered the 80s. This algorithm (commonly called Luby's algorithm) computes an MIS in $O(\log n)$ rounds (with high probability). This situation changed when Lenzen and Wattenhofer (PODC 2011) presented a randomized $O(\sqrt{\log n}\cdot \log\log n)$-round MIS algorithm for unoriented trees. This algorithm was improved by Barenboim et al. (FOCS 2012), resulting in an $O(\sqrt{\log n \cdot \log\log n})$-round MIS algorithm. The analyses of these tree MIS algorithms depends on "near independence" of probabilistic events, a feature of the tree structure of the network. In their paper, Lenzen and Wattenhofer hope that their algorithm and analysis could be extended to graphs with bounded arboricity. We show how to do this. By using a new tail inequality for read-k families of random variables due to Gavinsky et al. (Random Struct Algorithms, 2015), we show how to deal with dependencies induced by the recent tree MIS algorithms when they are executed on bounded arboricity graphs. Specifically, we analyze a version of the tree MIS algorithm of Barenboim et al. and show that it runs in $O(\mbox{poly}(\alpha) \cdot \sqrt{\log n \cdot \log\log n})$ rounds in the $\mathcal{CONGEST}$ model for graphs with arboricity $\alpha$. While the main thrust of this paper is the new probabilistic analysis via read-$k$ inequalities, for small values of $\alpha$, this algorithm is faster than the bounded arboricity MIS algorithm of Barenboim et al. We also note that recently (SODA 2016), Gaffari presented a novel MIS algorithm for general graphs that runs in $O(\log \Delta) + 2^{O(\sqrt{\log\log n})}$ rounds; a corollary of this algorithm is an $O(\log \alpha + \sqrt{\log n})$-round MIS algorithm on arboricity-$\alpha$ graphs.