An optimal bit complexity randomized distributed MIS algorithm

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 anonymou...

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Bibliographic Details
Published in:Distributed computing Vol. 23; no. 5-6; pp. 331 - 340
Main Authors: Métivier, Y., Robson, J. M., Saheb-Djahromi, N., Zemmari, A.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.04.2011
Springer Nature B.V
Springer Verlag
Series:Lecture notes in computer science
Subjects:
Algorithms
Channels
Complexity
Computer Communication Networks
Computer Hardware
Computer Science
Computer Systems Organization and Communication Networks
Distributed, Parallel, and Cluster Computing
Exchange
Graphs
Lectures
Management information systems
Messages
Software Engineering/Programming and Operating Systems
Theory of Computation
Neighbour Degree
Arbitrary Graph
Distant Vertex
Active Neighbour
Asymptotic Independence
bit complexity
Distributed graph algorithm
randomized algorithm
maximal independent set
ISSN:0178-2770, 1432-0452
Online Access:Get full text
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Summary: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 ).
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ISSN:0178-2770
1432-0452
DOI:10.1007/s00446-010-0121-5