# An optimal bit complexity randomized distributed MIS algorithm

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## Abstract

We present a randomized distributed maximal independent set (MIS) algorithm for arbitrary graphs of size *n* that halts in time *O*(log *n*) with probability 1 − *o*(*n* ^{−1}), and only needs messages containing 1 bit. Thus, its bit complexity par channel is *O*(log *n*). We assume that the graph is anonymous: unique identities are not available to distinguish between the processes; we only assume that each vertex distinguishes between its neighbours by locally known channel names. Furthermore we do not assume that the size (or an upper bound on the size) of the graph is known. This algorithm is optimal (modulo a multiplicative constant) for the bit complexity and improves the best previous randomized distributed MIS algorithms (deduced from the randomized PRAM algorithm due to Luby (SIAM J. Comput. 15:1036–1053, 1986)) for general graphs which is *O*(log^{2} * n*) per channel (it halts in time *O*(log *n*) and the size of each message is log *n*). This result is based on a powerful and general technique for converting unrealistic exchanges of messages containing real numbers drawn at random on each vertex of a network into exchanges of bits. Then we consider a natural question: what is the impact of a vertex inclusion in the MIS on distant vertices? We prove that this impact vanishes rapidly as the distance grows for bounded-degree vertices and we provide a counter-example that shows this result does not hold in general. We prove also that these results remain valid for Luby’s algorithm presented by Lynch (Distributed algorithms. Morgan Kaufman 1996) and by Wattenhofer (http://dcg.ethz.ch/lectures/fs08/distcomp/lecture/chapter4.pdf, 2007). This question remains open for the variant given by Peleg (Distributed computing—a locality-sensitive approach 2000).

## Keywords

Arbitrary Graph Active Neighbour Asymptotic Independence Distant Vertex Neighbour Degree## Preview

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