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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| Vydáno v: | Distributed computing Ročník 23; číslo 5-6; s. 331 - 340 |
|---|---|
| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Berlin/Heidelberg
Springer-Verlag
01.04.2011
Springer Nature B.V Springer Verlag |
| Edice: | Lecture notes in computer science |
| Témata: | |
| 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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| Issue | 5-6 |
| Keywords | Neighbour Degree Arbitrary Graph Distant Vertex Active Neighbour Asymptotic Independence bit complexity Distributed graph algorithm randomized algorithm maximal independent set |
| Language | English |
| License | http://www.springer.com/tdm Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0 |
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| References | Chalopin, Métivier (CR4) 2007; 80 Luby (CR15) 1986; 15 CR2 Moscibroda, Wattenhofer (CR17) 2008; 21 CR5 CR8 CR19 CR7 Alon, Babai, Itai (CR1) 1986; 7 CR9 CR16 Linial (CR14) 1992; 21 CR12 CR23 CR11 CR22 CR10 Ghosh (CR6) 2006 Bodlaender, Moran, Warmuth (CR3) 1994; 114 Santoro (CR20) 2007 Naor, Stockmeyer (CR18) 1995; 24 Kushilevitz, Nisan (CR13) 1999 Tel (CR21) 2000 H.L. Bodlaender (121_CR3) 1994; 114 E. Kushilevitz (121_CR13) 1999 M. Naor (121_CR18) 1995; 24 G. Tel (121_CR21) 2000 J. Chalopin (121_CR4) 2007; 80 N. Alon (121_CR1) 1986; 7 121_CR16 N. Linial (121_CR14) 1992; 21 121_CR19 T. Moscibroda (121_CR17) 2008; 21 121_CR10 121_CR5 N. Santoro (121_CR20) 2007 121_CR2 121_CR12 121_CR23 121_CR11 121_CR22 121_CR8 S. Ghosh (121_CR6) 2006 121_CR9 121_CR7 M. Luby (121_CR15) 1986; 15 |
| References_xml | – year: 2006 ident: CR6 publication-title: Distributed Systems—An Algorithmic Approach doi: 10.1201/9781420010848 – ident: CR22 – ident: CR19 – ident: CR2 – volume: 15 start-page: 1036 year: 1986 end-page: 1053 ident: CR15 article-title: A simple parallel algorithm for the maximal independent set problem publication-title: SIAM J. Comput. doi: 10.1137/0215074 – ident: CR16 – ident: CR12 – volume: 7 start-page: 567 issue: 4 year: 1986 end-page: 583 ident: CR1 article-title: A fast and simple randomized parallel algorithm for the maximal independent set publication-title: J. Algorithms doi: 10.1016/0196-6774(86)90019-2 – volume: 24 start-page: 1259 issue: 6 year: 1995 end-page: 1277 ident: CR18 article-title: What can be computed locally? publication-title: SIAM J. Comput. doi: 10.1137/S0097539793254571 – year: 2007 ident: CR20 publication-title: Design and Analysis of Distributed Algorithms – ident: CR10 – ident: CR11 – ident: CR9 – volume: 21 start-page: 193 year: 1992 end-page: 201 ident: CR14 article-title: Locality in distributed graph algorithms publication-title: SIAM J. Comput. doi: 10.1137/0221015 – year: 1999 ident: CR13 publication-title: Communication Complexity – volume: 21 start-page: 271 issue: 4 year: 2008 end-page: 284 ident: CR17 article-title: Coloring unstructured radio networks publication-title: Distrib. Comput. doi: 10.1007/s00446-008-0070-4 – ident: CR5 – year: 2000 ident: CR21 publication-title: Introduction to Distributed Algorithms – ident: CR7 – ident: CR8 – volume: 114 start-page: 34 issue: 2 year: 1994 end-page: 50 ident: CR3 article-title: The distributed bit complexity of the ring: from the anonymous case to the non-anonymous case publication-title: Inf. comput. doi: 10.1006/inco.1994.1002 – ident: CR23 – volume: 80 start-page: 221 issue: 1-3 year: 2007 end-page: 246 ident: CR4 article-title: An efficient message passing election algorithm based on mazurkiewicz’s algorithm publication-title: Fundam. Inform. – ident: 121_CR2 doi: 10.1109/SFCS.1989.63504 – volume-title: Design and Analysis of Distributed Algorithms year: 2007 ident: 121_CR20 – volume: 114 start-page: 34 issue: 2 year: 1994 ident: 121_CR3 publication-title: Inf. comput. doi: 10.1006/inco.1994.1002 – ident: 121_CR12 doi: 10.1145/1146381.1146387 – volume: 21 start-page: 271 issue: 4 year: 2008 ident: 121_CR17 publication-title: Distrib. Comput. doi: 10.1007/s00446-008-0070-4 – ident: 121_CR7 doi: 10.1007/978-3-662-12788-9 – ident: 121_CR16 – ident: 121_CR23 doi: 10.1145/800135.804414 – volume: 21 start-page: 193 year: 1992 ident: 121_CR14 publication-title: SIAM J. Comput. doi: 10.1137/0221015 – volume: 7 start-page: 567 issue: 4 year: 1986 ident: 121_CR1 publication-title: J. Algorithms doi: 10.1016/0196-6774(86)90019-2 – ident: 121_CR10 doi: 10.1007/11561927_21 – ident: 121_CR19 doi: 10.1137/1.9780898719772 – volume-title: Introduction to Distributed Algorithms year: 2000 ident: 121_CR21 doi: 10.1017/CBO9781139168724 – volume: 80 start-page: 221 issue: 1-3 year: 2007 ident: 121_CR4 publication-title: Fundam. Inform. – ident: 121_CR9 – ident: 121_CR5 doi: 10.1145/1326554.1326557 – ident: 121_CR8 doi: 10.1145/800057.808690 – ident: 121_CR22 – volume: 24 start-page: 1259 issue: 6 year: 1995 ident: 121_CR18 publication-title: SIAM J. Comput. doi: 10.1137/S0097539793254571 – ident: 121_CR11 doi: 10.1145/1011767.1011811 – volume-title: Communication Complexity year: 1999 ident: 121_CR13 – volume: 15 start-page: 1036 year: 1986 ident: 121_CR15 publication-title: SIAM J. Comput. doi: 10.1137/0215074 – volume-title: Distributed Systems—An Algorithmic Approach year: 2006 ident: 121_CR6 doi: 10.1201/9781420010848 |
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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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| 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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