Skip to main content
SpringerLink
Log in

Distributed Symmetry Breaking in Hypergraphs

  • Conference paper
Distributed Computing (DISC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8784))

Included in the following conference series:

Abstract

Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and Δ + 1-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in hypergraphs. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs.

In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in O(log2 n) rounds (n is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper — an O(Δεpolylog n)-round hypergraph MIS algorithm in the CONGEST model where Δ is the maximum node degree of the hypergraph and ε > 0 is any arbitrarily small constant. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.

To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the balanced minimal dominating set problem (left open in Harris et al. [ICALP 2013]) and the minimal connected dominating set problem.

Our work shows that while some local symmetry breaking problems such as coloring can be solved in polylogarithmic rounds in both the LOCAL and CONGEST models, for many other hypergraph problems such as MIS, hitting set, and maximal clique, it remains challenging to obtain polylogarithmic time algorithms in the CONGEST model. This work is a step towards understanding this dichotomy in the complexity of hypergraph problems as well as using hypergraphs to design fast distributed algorithms for problems in (standard) graphs.

The full version is available at http://arxiv.org/abs/1405.1649

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alon, N., Spencer, J.: The Probabilistic Method. Series in Discrete Mathematics and Optimization. Wiley (2008), http://books.google.co.jp/books?id=V8YgNioxF6AC

  2. Avin, C., Lando, Y., Lotker, Z.: Radio cover time in hyper-graphs. Ad Hoc Networks 12, 278–290 (2014)

    Article  Google Scholar 

  3. Azar, Y., Naor, J., Rom, R.: The competitiveness of on-line assignments. In: Frederickson, G.N. (ed.) SODA, pp. 203–210. ACM/SIAM (1992), http://dblp.uni-trier.de/db/conf/soda/soda92.html#AzarNR92

  4. Balakrishnan, H., Barrett, C.L., Kumar, V.S.A., Marathe, M.V., Thite, S.: The distance-2 matching problem and its relationship to the mac-layer capacity of ad hoc wireless networks. IEEE Journal on Selected Areas in Communications 22(6), 1069–1079 (2004)

    Article  Google Scholar 

  5. Barenboim, L., Elkin, M.: Distributed Graph Coloring: Fundamentals and Recent Developments. Synthesis Lectures on Distributed Computing Theory. Morgan & Claypool Publishers (2013)

    Google Scholar 

  6. Beame, P., Luby, M.: Parallel search for maximal independence given minimal dependence. In: SODA, pp. 212–218 (1990)

    Google Scholar 

  7. Bercea, I., Goyal, N., Harris, D., Srinivasan, A.: On computing maximal independent sets of hypergraphs in parallel. In: SPAA (June 2014)

    Google Scholar 

  8. Chen, Y.P., Liestman, A.L.: Approximating minimum size weakly-connected dominating sets for clustering mobile ad hoc networks. In: MobiHoc, pp. 165–172 (2002)

    Google Scholar 

  9. Chlamtac, I., Kutten, S.: Tree-based broadcasting in multihop radio networks. IEEE Trans. Computers 36(10), 1209–1223 (1987)

    Article  Google Scholar 

  10. Dai, F., Wu, J.: An extended localized algorithm for connected dominating set formation in ad hoc wireless networks. IEEE Trans. Parallel Distrib. Syst. 15(10), 908–920 (2004), http://dx.doi.org/10.1109/TPDS.2004.48

    Article  Google Scholar 

  11. Das, B., Bharghavan, V.: Routing in ad-hoc networks using minimum connected dominating sets. In: ICC (1), pp. 376–380 (1997)

    Google Scholar 

  12. Dubhashi, D.P., Mei, A., Panconesi, A., Radhakrishnan, J., Srinivasan, A.: Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons. J. Comput. Syst. Sci. 71(4), 467–479 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ephremides, A., Truong, T.V.: Scheduling broadcasts in multihop radio networks. IEEE Transactions on Communications 38(4), 456–460 (1990)

    Article  Google Scholar 

  14. Garay, J.A., Kutten, S., Peleg, D.: A sublinear time distributed algorithm for minimum-weight spanning trees. SIAM J. Comput. 27(1), 302–316 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghaffari, M.: Near-optimal distributed approximation of minimum-weight connected dominating set. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014, Part II. LNCS, vol. 8573, pp. 483–494. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  16. Harris, D.G., Morsy, E., Pandurangan, G., Robinson, P., Srinivasan, A.: Efficient computation of balanced structures. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 581–593. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  17. Karp, R.M., Ramachandran, V.: Parallel algorithms for shared-memory machines. In: Handbook of Theoretical Computer Science, vol. A, pp. 869–942 (1990)

    Google Scholar 

  18. Karp, R.M., Upfal, E., Wigderson, A.: The complexity of parallel search. J. Comput. Syst. Sci. 36(2), 225–253 (1988), announced at STOC 1985 and FOCS 1985

    Google Scholar 

  19. Kelsen, P.: On the parallel complexity of computing a maximal independent set in a hypergraph. In: STOC, pp. 339–350 (1992)

    Google Scholar 

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

    Google Scholar 

  21. Kutten, S., Pandurangan, G., Peleg, D., Robinson, P., Trehan, A.: On the complexity of universal leader election. In: PODC, pp. 100–109 (2013)

    Google Scholar 

  22. Kutten, S., Peleg, D.: Fast distributed construction of small k-dominating sets and applications. J. Algorithms 28(1), 40–66 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Linial, N.: Dijkstra award talk Jerusalem (2013), http://www.cs.huji.ac.il/~nati/PAPERS/disc_2013.pdf

  25. Linial, N., Saks, M.E.: Low diameter graph decompositions. Combinatorica 13(4), 441–454 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15(4), 1036–1053 (1986), announced at STOC 1985

    Google Scholar 

  27. Luczak, T., Szymanska, E.: A parallel randomized algorithm for finding a maximal independent set in a linear hypergraph. J. Algorithms 25(2), 311–320 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Makino, K., Kameda, T.: Efficient generation of all regular non-dominated coteries. In: PODC, pp. 279–288 (2000)

    Google Scholar 

  29. Métivier, Y., Robson, J.M., Saheb-Djahromi, N., Zemmari, A.: An optimal bit complexity randomized distributed mis algorithm. Distributed Computing 23(5-6), 331–340 (2011)

    Article  MATH  Google Scholar 

  30. Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM, Philadelphia (2000)

    Google Scholar 

  31. Sarma, A.D., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G., Peleg, D., Wattenhofer, R.: Distributed verification and hardness of distributed approximation. SIAM J. Comput. 41(5), 1235–1265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Thurimella, R.: Sub-linear distributed algorithms for sparse certificates and biconnected components. J. Algorithms 23(1), 160–179 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wasserman, S., Faust, K.: Social network analysis: Methods and Applications. Cambridge University Press (1994)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of IE&M, Technion, Haifa, Israel

    Shay Kutten

  2. Faculty of Computer Science, University of Vienna, Austria

    Danupon Nanongkai

  3. Department of Computer Science, University of Houston, USA

    Gopal Pandurangan

  4. School of Computing, National University of Singapore, Singapore

    Peter Robinson

Authors
  1. Shay Kutten
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Danupon Nanongkai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gopal Pandurangan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Peter Robinson
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Freiburg, 79110, Freiburg, Germany

    Fabian Kuhn

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kutten, S., Nanongkai, D., Pandurangan, G., Robinson, P. (2014). Distributed Symmetry Breaking in Hypergraphs. In: Kuhn, F. (eds) Distributed Computing. DISC 2014. Lecture Notes in Computer Science, vol 8784. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45174-8_32

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-45174-8_32

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-45173-1

  • Online ISBN: 978-3-662-45174-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation