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(logn) rounds (with high probability). This situation changed
when Lenzen and Wattenhofer (PODC 2011) presented a randomized O(logn⋅loglogn)-round MIS algorithm for unoriented trees. This algorithm
was improved by Barenboim et al. (FOCS 2012), resulting in an O(logn⋅loglogn)-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 CONGEST model for
graphs with arboricity α.
While the main thrust of this paper is the new probabilistic analysis via
read-k inequalities, for small values of α, 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(logΔ)+2O(loglogn) rounds; a
corollary of this algorithm is an O(logα+logn)-round MIS
algorithm on arboricity-α graphs.Comment: To appear in PODC 2016 as a brief announcemen