Using Read-k Inequalities to Analyze a Distributed MIS Algorithm

Abstract

Until recently, the fastest distributed MIS algorithm, even for simple graph classes such as unoriented trees that can contain large independent sets within neighborhoods, has been the simple randomized algorithm discovered independently by several researchers in the late 80s. This algorithm (commonly called Luby’s algorithm) computes an MIS of an n-node graph in O(log n) communication rounds (with high probability). This situation changed when Lenzen and Wattenhofer (PODC 2011) presented a distributed (randomized) MIS algorithm for unoriented treesrunning in O( sqrt (log n * log log n)) rounds. This algorithm was slightly improved by Barenboim et al. (FOCS 2012), resulting in an O( sqrt (log n * log log n))-round (randomized) MIS algorithm for trees. At their core, these algorithms still run Luby's algorithm, but only up to the point at which the graph has been "shattered" into small connected components that can be independently processed in parallel. 

The analyses of these tree MIS algorithms critically depends on "near independence" among probabilistic events, a feature that arises from the tree structure of the network. In their paper, Lenzen and Wattenhofer express hope that their algorithm and analysis could be extended to graphs with bounded arboricity. We show how to do this in the current paper. 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(poly(a) * sqrt ( log n * log log n)) rounds in the CONGEST model for graphs with arboricity a.

While the main thrust of this paper is the new probabilistic analysis via read-k inequalities, we point out that for small values of a, this algorithm is faster than the MIS algorithm of Barenboim et al. specifically designed for bounded arboricity graphs. In this context, it should be noted that recently (in SODA 2016) Ghaffari presented a novel distributed MIS algorithm for general graphs that runs in O (log d) + 2^O(sqrt(log log n)) rounds and a corollary of this algorithm is an O(log d + sqrt (log n))-round MIS algorithm on graphs with arboricity a.