research-article

The Locality of Distributed Symmetry Breaking

Authors:

Leonid Barenboim,

Johannes SchneiderAuthors Info & Claims

Journal of the ACM (JACM), Volume 63, Issue 3

Article No.: 20, Pages 1 - 45

https://doi.org/10.1145/2903137

Published: 28 June 2016 Publication History

Abstract

Symmetry-breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this article we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry-breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes the following:

—An MIS algorithm running in O(log²Δ + 2^{o(√log log n)}) time, where Δ is the maximum degree. This is the first MIS algorithm to improve on the 1986 algorithms of Luby and Alon, Babai, and Itai, when log n ≪ Δ ≪ 2√log n, and comes close to the Ω(log Δ / log log Δ lower bound of Kuhn, Moscibroda, and Wattenhofer.

—A maximal matching algorithm running in O(log Δ + log ⁴log n) time. This is the first significant improvement to the 1986 algorithm of Israeli and Itai. Moreover, its dependence on Δ is nearly optimal.

—A (Δ + 1)-coloring algorithm requiring O(log Δ + 2^{o(√log log n)} time, improving on an O(log Δ + √log n)-time algorithm of Schneider and Wattenhofer.

—A method for reducing symmetry-breaking problems in low arboricity/degeneracy graphs to low-degree graphs. (Roughly speaking, the arboricity or degeneracy of a graph bounds the density of any subgraph.) Corollaries of this reduction include an O(√log n)-time maximal matching algorithm for graphs with arboricity up to 2√log n and an O(log ^2/3n)-time MIS algorithm for graphs with arboricity up to 2^{(log n)1/3}.

Each of our algorithms is based on a simple but powerful technique for reducing a randomized symmetry-breaking task to a corresponding deterministic one on a poly(log n)-size graph.

References

[1]

N. Alon. 1991. A parallel algorithmic version of the local lemma. Rand. Struct. Algor. 2, 4 (1991), 367--378.

Digital Library

[2]

N. Alon, L. Babai, and A. Itai. 1986. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algor. 7 (1986), 567--583.

Digital Library

[3]

A. Amir, O. Kapah, T. Kopelowitz, M. Naor, and E. Porat. 2014. The family holiday gathering problem or fair and periodic scheduling of independent sets. CoRR abs/1408.2279 (2014).

[4]

B. Awerbuch, A. V. Goldberg, M. Luby, and S. A. Plotkin. 1989. Network decomposition and locality in distributed computation. In Proceedings 30th IEEE Symposium on Foundations of Computer Science (FOCS). 364--369.

Digital Library

[5]

R. Bar-Yehuda, K. Censor-Hillel, and G. Schwartzman. 2016. A distributed (2 + &epsi;)-approximation for vertex cover in O(log Δ/&epsi;log log Δ) rounds. CoRR abs/1602.03713 (2016).

[6]

L. Barenboim. 2015. Deterministic (Δ + 1)-coloring in sublinear (in Δ) time in static, dynamic and faulty networks. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing (PODC). 345--354.

Digital Library

[7]

L. Barenboim and M. Elkin. 2010. Sublogarithmic distributed MIS algorithm for sparse graphs using Nash-Williams decomposition. Distrib. Comput. 22, 5--6 (2010), 363--379.

Digital Library

[8]

L. Barenboim and M. Elkin. 2011. Deterministic distributed vertex coloring in polylogarithmic time. J. ACM 58, 5 (2011), 23.

Digital Library

[9]

L. Barenboim and M. Elkin. 2013. Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers, San Francisco, CA.

Digital Library

[10]

L. Barenboim, M. Elkin, and F. Kuhn. 2014. Distributed (Δ + 1)-coloring in linear (in Δ) time. SIAM J. Comput. 43, 1 (2014), 72--95.

Digital Library

[11]

J. Beck. 1991. An algorithmic approach to the Lovász local lemma. I. Rand. Struct. Algor. 2, 4 (1991), 343--366.

Digital Library

[12]

T. Bisht, K. Kothapalli, and S. V. Pemmaraju. 2014. Brief announcement: Super-fast t-ruling sets. In Proceedings 33rd ACM Symposium on Principles of Distributed Computing (PODC). 379--381.

Digital Library

[13]

K.-M. Chung, S. Pettie, and H.-H. Su. 2014. Distributed algorithms for the Lovász local lemma and graph coloring. In Proceedings 33rd ACM Symposium on Principles of Distributed Computing (PODC). 134--143.

Digital Library

[14]

R. Cole and U. Vishkin. 1986. Deterministic coin tossing with applications to optimal parallel list ranking. Inform. Control 70, 1 (1986), 32--53.

Digital Library

[15]

D. P. Dubhashi and A. Panconesi. 2009. Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, Cambridge.

Digital Library

[16]

D. P. Dubhashi and D. Ranjan. 1998. Balls and bins: A study in negative dependence. J. Rand. Struct. Algs. 13, 2 (1998), 99--124.

Digital Library

[17]

M. Elkin, S. Pettie, and H. H. Su. 2015. (2Δ − 1)-edge coloring is much easier than maximal matching in the distributed setting. In Proceedings 26th ACM-SIAM Symposium on Discrete Algorithms (SODA). 355--370.

Digital Library

[18]

B. Gfeller and E. Vicari. 2007. A randomized distributed algorithm for the maximal independent set problem in growth-bounded graphs. In Proceedings 26th Annual ACM Symposium on Principles of Distributed Computing (PODC). 53--60.

Digital Library

[19]

M. Ghaffari. 2016. An improved distributed algorithm for maximal independent set. In Proceedings 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). 270--277.

Digital Library

[20]

M. Hańćkowiak, M. Karoński, and A. Panconesi. 2001. On the distributed complexity of computing maximal matchings. SIAM J. Discr. Math. 15, 1 (2001), 41--57 (electronic).

Digital Library

[21]

D. Harris, J. Schneider, and H.-H. Su. 2016. Distributed (Δ + 1)-coloring in sublogarithmic rounds. In Proceedings 48th ACM Symposium on Theory of Computing (STOC).

Digital Library

[22]

A. Israeli and A. Itai. 1986. A fast and simple randomized parallel algorithm for maximal matching. Info. Proc. Lett. 22, 2 (1986), 77--80.

Digital Library

[23]

Ö. Johansson. 1999. Simple distributed Δ + 1-coloring of graphs. Info. Proc. Lett. 70, 5 (1999), 229--232.

Digital Library

[24]

A. Korman, J.-S. Sereni, and L. Viennot. 2013. Toward more localized local algorithms: Removing assumptions concerning global knowledge. Distrib. Comput. 26, 5--6 (2013), 289--308.

Digital Library

[25]

K. Kothapalli and S. V. Pemmaraju. 2011. Distributed graph coloring in a few rounds. In Proceedings 30th Annual ACM Symposium on Principles of Distributed Computing (PODC). 31--40.

Digital Library

[26]

K. Kothapalli and S. V. Pemmaraju. 2012. Super-fast 3-ruling sets. In Proceedings IARCS Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS). LIPIcs, Vol. 18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 136--147.

[27]

K. Kothapalli, C. Scheideler, M. Onus, and C. Schindelhauer. 2006. Distributed coloring in O&tilde;(√log n) bit rounds. In Proceedings 20th International Parallel and Distributed Processing Symposium (IPDPS).

Digital Library

[28]

F. Kuhn, T. Moscibroda, and R. Wattenhofer. 2004. What cannot be computed locally! In Proceedings 23rd Annual ACM Symposium on Principles of Distributed Computing (PODC). 300--309.

Digital Library

[29]

F. Kuhn, T. Moscibroda, and R. Wattenhofer. 2010. Local computation: Lower and upper bounds. CoRR abs/1011.5470 (2010).

[30]

F. Kuhn and R. Wattenhofer. 2006. On the complexity of distributed graph coloring. In Proceedings 25th Annual ACM Symposium on Principles of Distributed Computing (PODC). 7--15.

Digital Library

[31]

C. Lenzen and R. Wattenhofer. 2011. MIS on trees. In Proceedings 30th Annual ACM Symposium on Principles of Distributed Computing (PODC). 41--48.

Digital Library

[32]

N. Linial. 1992. Locality in distributed graph algorithms. SIAM J. Comput. 21, 1 (1992), 193--201.

Digital Library

[33]

N. Linial and M. E. Saks. 1993. Low diameter graph decompositions. Combinatorica 13, 4 (1993), 441--454.

[34]

M. Luby. 1986. A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15, 4 (1986), 1036--1053.

Digital Library

[35]

R. A. Moser and G. Tardos. 2010. A constructive proof of the general Lovász local lemma. J. ACM 57, 2 (2010).

Digital Library

[36]

C. St. J. A. Nash-Williams. 1964. Decomposition of finite graphs into forests. J. London Math. Soc. 39 (1964), 12.

[37]

A. Panconesi and R. Rizzi. 2001. Some simple distributed algorithms for sparse networks. Distrib. Comput. 14, 2 (2001), 97--100.

Digital Library

[38]

A. Panconesi and A. Srinivasan. 1996. On the complexity of distributed network decomposition. J. Algor. 20, 2 (1996), 356--374.

Digital Library

[39]

D. Peleg. 2000. Distributed Computing: A Locality-Sensitive Approach. SIAM, Philadelphia, PA.

Digital Library

[40]

S. Pettie and H.-H. Su. 2015. Distributed algorithms for coloring triangle-free graphs. Inform. Comput. 243 (2015), 263--280.

Digital Library

[41]

R. Rubinfeld, G. Tamir, S. Vardi, and N. Xie. 2011. Fast local computation algorithms. In Proceedings of the 1st Symposium on Innovations in Computer Science (ICS). 223--238. See also CoRR abs/1104.1377.

[42]

J. Schneider, M. Elkin, and R. Wattenhofer. 2013. Symmetry breaking depending on the chromatic number or the neighborhood growth. Theor. Comput. Sci. 509 (2013), 40--50.

Digital Library

[43]

J. Schneider and R. Wattenhofer. 2010a. A new technique for distributed symmetry breaking. In Proceedings 29th Annual ACM Symposium on Principles of Distributed Computing (PODC). 257--266.

Digital Library

[44]

J. Schneider and R. Wattenhofer. 2010b. An optimal maximal independent set algorithm for bounded-independence graphs. Distrib. Comput. 22, 5--6 (2010), 349--361.

Digital Library

[45]

H.-H. Su. 2015. Algorithms for Fundamental Problems in Computer Networks. Ph.D. Dissertation. University of Michigan.

[46]

M. Szegedy and S. Vishwanathan. 1993. Locality based graph coloring. In Proceedings 25th ACM Symposium on Theory of Computing (STOC). 201--207.

Digital Library

Cited By

Giakkoupis GTurau VZiccardi I(2025)Luby's MIS algorithms made self-stabilizingInformation Processing Letters10.1016/j.ipl.2024.106531188(106531)Online publication date: Feb-2025
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Giliberti JParsaeian ZKuznetsov PGelles ROlivetti D(2024)Brief Announcement: Massively Parallel Ruling Set Made DeterministicProceedings of the 43rd ACM Symposium on Principles of Distributed Computing10.1145/3662158.3662816(523-526)Online publication date: 17-Jun-2024
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Fuchs MKuhn FKuznetsov PGelles ROlivetti D(2024)Brief Announcement: Simpler and More General Distributed Coloring Based on Simple List Defective Coloring AlgorithmsProceedings of the 43rd ACM Symposium on Principles of Distributed Computing10.1145/3662158.3662808(425-428)Online publication date: 17-Jun-2024
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Index Terms

The Locality of Distributed Symmetry Breaking
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
2. Theory of computation
  1. Design and analysis of algorithms

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cover image Journal of the ACM

Journal of the ACM Volume 63, Issue 3

September 2016

303 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2957788

Editor:
Éva Tardos
Cornell University

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Copyright © 2016 ACM.

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Publication History

Published: 28 June 2016

Accepted: 01 March 2016

Revised: 01 February 2016

Received: 01 February 2015

Published in JACM Volume 63, Issue 3

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Israel Science Foundation
Israeli Academy of Science
US-Israel Binational Science Foundation
NSF
Center for Massive Data Algorithmics (MADALGO)
Aarhus University
Danish National Research Foundation

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Cited By

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https://doi.org/10.1016/j.ipl.2024.106531
Giliberti JParsaeian ZKuznetsov PGelles ROlivetti D(2024)Brief Announcement: Massively Parallel Ruling Set Made DeterministicProceedings of the 43rd ACM Symposium on Principles of Distributed Computing10.1145/3662158.3662816(523-526)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3662158.3662816
Fuchs MKuhn FKuznetsov PGelles ROlivetti D(2024)Brief Announcement: Simpler and More General Distributed Coloring Based on Simple List Defective Coloring AlgorithmsProceedings of the 43rd ACM Symposium on Principles of Distributed Computing10.1145/3662158.3662808(425-428)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3662158.3662808
Flin MMittal PKuznetsov PGelles ROlivetti D(2024)(Δ+1) Vertex Coloring in O(n) CommunicationProceedings of the 43rd ACM Symposium on Principles of Distributed Computing10.1145/3662158.3662796(416-424)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3662158.3662796
Ghaffari MGrunau CMohar BShinkar IO'Donnell R(2024)Dynamic O(Arboricity) Coloring in Polylogarithmic Worst-Case TimeProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649782(1184-1191)Online publication date: 10-Jun-2024
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Dahal SSuomela J(2024)Distributed half-integral matching and beyondTheoretical Computer Science10.1016/j.tcs.2023.114278982:COnline publication date: 8-Jan-2024
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DONG RIZUMI TKITAMURA NSUDO YMASUZAWA T(2023)Loosely-Stabilizing Algorithm on Almost Maximal Independent SetIEICE Transactions on Information and Systems10.1587/transinf.2023EDP7075E106.D:11(1762-1771)Online publication date: 1-Nov-2023
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Maus Y(2023)Distributed Graph Coloring Made EasyACM Transactions on Parallel Computing10.1145/360589610:4(1-21)Online publication date: 17-Aug-2023
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Giakkoupis GZiccardi IOshman RNolin AHalldorsson MBalliu A(2023)Distributed Self-Stabilizing MIS with Few States and Weak CommunicationProceedings of the 2023 ACM Symposium on Principles of Distributed Computing10.1145/3583668.3594581(310-320)Online publication date: 19-Jun-2023
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Maus YPeltonen SUitto JOshman RNolin AHalldorsson MBalliu A(2023)Distributed Symmetry Breaking on Power Graphs via SparsificationProceedings of the 2023 ACM Symposium on Principles of Distributed Computing10.1145/3583668.3594579(157-167)Online publication date: 19-Jun-2023
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