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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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
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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 ).
AbstractList	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(log2 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). We present a randomised distributed maximal independent set (MIS) algorithm for arbitrary graphs of size $n$ that halts in time $O(\log n)$ with probability $1-o(n^{-1})$, each message containing $1$ bit: thus its bit complexity per channel is $O(\log n)$ (the bit complexity is the number of bits we need to solve a distributed task, it measures the communication complexity). We assume that the graph is anonymous: unique identities are not available to distinguish 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 randomised distributed MIS algorithms (deduced from the randomised PRAM algorithm due to Luby) 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. 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 and by Wattenhofer. This question remains open for the variant given by Peleg. 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 ). 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 a o(n super(a1)), 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. 15:1036a1053, 1986)) for general graphs which is O(log super(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 Lubyas 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 computingaa locality-sensitive approach 2000). 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 ^sup -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^sup 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).[PUBLICATION ABSTRACT]
Author	Métivier, Y. Zemmari, A. Saheb-Djahromi, N. Robson, J. M.
Author_xml	– sequence: 1 givenname: Y. surname: Métivier fullname: Métivier, Y. organization: Université de Bordeaux, LaBRI UMR CNRS 5800 – sequence: 2 givenname: J. M. surname: Robson fullname: Robson, J. M. organization: Université de Bordeaux, LaBRI UMR CNRS 5800 – sequence: 3 givenname: N. surname: Saheb-Djahromi fullname: Saheb-Djahromi, N. organization: Université de Bordeaux, LaBRI UMR CNRS 5800 – sequence: 4 givenname: A. surname: Zemmari fullname: Zemmari, A. email: zemmari@labri.fr organization: Université de Bordeaux, LaBRI UMR CNRS 5800
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Cites_doi	10.1201/9781420010848 10.1137/0215074 10.1016/0196-6774(86)90019-2 10.1137/S0097539793254571 10.1137/0221015 10.1007/s00446-008-0070-4 10.1006/inco.1994.1002 10.1109/SFCS.1989.63504 10.1145/1146381.1146387 10.1007/978-3-662-12788-9 10.1145/800135.804414 10.1007/11561927_21 10.1137/1.9780898719772 10.1017/CBO9781139168724 10.1145/1326554.1326557 10.1145/800057.808690 10.1145/1011767.1011811
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Keywords	Neighbour Degree Arbitrary Graph Distant Vertex Active Neighbour Asymptotic Independence bit complexity Distributed graph algorithm randomized algorithm maximal independent set
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Snippet	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... 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... 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 a o(n... We present a randomised distributed maximal independent set (MIS) algorithm for arbitrary graphs of size $n$ that halts in time $O(\log n)$ with probability...
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StartPage	331
SubjectTerms	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
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Title	An optimal bit complexity randomized distributed MIS algorithm
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Volume	23
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