Distributed Computing

, Volume 26, Issue 4, pp 195–208

Beeping a maximal independent set

  • Yehuda Afek
  • Noga Alon
  • Ziv Bar-Joseph
  • Alejandro Cornejo
  • Bernhard Haeupler
  • Fabian Kuhn
Article

Abstract

We consider the problem of computing a maximal independent set (MIS) in an extremely harsh broadcast model that relies only on carrier sensing. The model consists of an anonymous broadcast network in which nodes have no knowledge about the topology of the network or even an upper bound on its size. Furthermore, it is assumed that an adversary chooses at which time slot each node wakes up. At each time slot a node can either beep, that is, emit a signal, or be silent. At a particular time slot, beeping nodes receive no feedback, while silent nodes can only differentiate between none of its neighbors beeping, or at least one of its neighbors beeping. We start by proving a lower bound that shows that in this model, it is not possible to locally converge to an MIS in sub-polynomial time. We then study four different relaxations of the model which allow us to circumvent the lower bound and find an MIS in polylogarithmic time. First, we show that if a polynomial upper bound on the network size is known, it is possible to find an MIS in \(\mathcal O (\log ^3 n)\) time. Second, if we assume sleeping nodes are awoken by neighboring beeps, then we can also find an MIS in \(\mathcal O (\log ^3 n)\) time. Third, if in addition to this wakeup assumption we allow sender-side collision detection, that is, beeping nodes can distinguish whether at least one neighboring node is beeping concurrently or not, we can find an MIS in \(\mathcal O (\log ^2 n)\) time. Finally, if instead we endow nodes with synchronous clocks, it is also possible to find an MIS in \(\mathcal O (\log ^2 n)\) time.

Keywords

Maximal independent set Distributed Beeps Radio networks Asynchronous wakeup 

References

  1. 1.
    Afek, Y., Alon, N., Barad, O., Hornstein, E., Barkai, N., Bar-Joseph, Z.: A biological solution to a fundamental distributed computing problem. Science 331(6014), 183–185 (2011)Google Scholar
  2. 2.
    Alon, N., Babai, L., Itai, A.: A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms 7(4), 567–583 (1986)Google Scholar
  3. 3.
    Awerbuch, B., Goldberg, A.V., Luby, M., Plotkin, S.A.: Network decomposition and locality in distributed computation. In: Proc. of the 30th Symposium on Foundations of Computer Science (FOCS), pp. 364–369 (1989)Google Scholar
  4. 4.
    Chlebus, B., Gasieniec, L., Gibbons, A., Pelc, A., Rytter, W.: Deterministic broadcasting in unknown radio networks. In: Prof. 11th ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 861–870 (2000)Google Scholar
  5. 5.
    Collier, J.R., Monk, N.A., Maini, P.K., Lewis, J.H.: Pattern formation by lateral inhibition with feedback: a mathematical model of delta-notch intercellular signalling. J. Theor. Biol. 183(4), 429–446 (1996)CrossRefGoogle Scholar
  6. 6.
    Cornejo, A., Kuhn, F.: Deploying wireless networks with beeps. In: Proc. 24th Symposium on Distributed Computing (DISC), pp. 148–162 (2010) Google Scholar
  7. 7.
    Degesys, J., Rose, I., Patel, A., Nagpal, R.: Desync: self-organizing desynchronization and TDMA on wireless sensor networks. In: Prof. 6th Conf. on Information Processing in Sensor Networks (IPSN), p. 20 (2007)Google Scholar
  8. 8.
    Flury, R., Wattenhofer, R.: Slotted programming for sensor networks. In: Proc. 9th Conference on Information Processing in Sensor Networks (IPSN) (2010)Google Scholar
  9. 9.
    Ilcinkas, D., Kowalski, D., Pelc, A.: Fast radio broadcasting with advice. Theor. Comput. Sci. 411, 14–15 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kuhn, F., Moscibroda, T., Nieberg, T., Wattenhofer, R.: Fast deterministic distributed maximal independent set computation on growth-bounded graphs. In: Proc. 19th International Symposium on, Distributed Computing (DISC’05), pp. 273–287 (2005)Google Scholar
  11. 11.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: What cannot be computed locally! In: Proc. 23rd ACM Symposium on Principles of Distributed Computing (PODC), pp. 300–309 (2004)Google Scholar
  12. 12.
    Kuhn, F., Moscibroda, T., Wattenhofer R.: The price of being near-sighted. In: Proc. 17th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 980–989 (2006)Google Scholar
  13. 13.
    Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15, 1036–1053 (1986)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Métivier, Y., Michael Robson, J., Saheb-Djahromi, N., Zemmari, A.: An optimal bit complexity randomized distributed mis algorithm. Distrib. Comput. 23, 331–340 (2011)Google Scholar
  15. 15.
    Moscibroda, T., Wattenhofer, R.: Efficient computation of maximal independent sets in structured multi-hop radio networks. In: Proc. of 1st International Conference on Mobile Ad Hoc Sensor Systems (MASS) (2004)Google Scholar
  16. 16.
    Moscibroda, T., Wattenhofer, R.: Maximal independent sets in radio networks. In: Proc. 24th Symposium on Principles of Distributed Computing (PODC) (2005)Google Scholar
  17. 17.
    Motskin, A., Roughgarden, T., Skraba, P., Guibas, L.: Lightweight coloring and desynchronization for networks. In: Proc. 28th IEEE Conf. on Computer Communications (INFOCOM) (2009)Google Scholar
  18. 18.
    Panconesi, A., Srinivasan, A.: On the complexity of distributed network decomposition. J. Algorithms 20(2), 581–592 (1995)MathSciNetGoogle Scholar
  19. 19.
    Peleg, D.: Distributed computing: a locality-sensitive approach. Society for Industrial and Applied Mathematics, Philadelphia (2000)MATHCrossRefGoogle Scholar
  20. 20.
    Scheideler, C., Richa, A., Santi, P.: An \({O}(\log n)\) dominating set protocol for wireless ad-hoc networks under the physical interference model. In: Proc. 9th Symposium on Mobile Ad Hoc Networking and Computing (MOBIHOC) (2008)Google Scholar
  21. 21.
    Schneider, J., Wattenhofer, R.: A Log-Star Maximal Independent Set Algorithm for Growth-Bounded Graphs. In: Proc. 28th Symposium on Principles of Distributed Computing (PODC) (2008)Google Scholar
  22. 22.
    Schneider, J., Wattenhofer, R.: What is the use of collision detection (in wireless networks)? In: Proc. of 24th Symposium on Distributed Computing (DISC) (2010)Google Scholar
  23. 23.
    Wan, P.J., Alzoubi, K.M., Frieder, O.: Distributed construction of connected dominating set in wireless ad hoc networks. Mobile Netw. Appl. 6343, 133–147 (2004)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Yehuda Afek
    • 1
  • Noga Alon
    • 2
  • Ziv Bar-Joseph
    • 3
  • Alejandro Cornejo
    • 4
  • Bernhard Haeupler
    • 4
  • Fabian Kuhn
    • 5
  1. 1.The Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Sackler School of MathematicsTel Aviv UniversityTel AvivIsrael
  3. 3.School of Computer ScienceCarnegie Mellon Univ.PittsburghUSA
  4. 4.Computer Science and Artificial Intelligence LaboratoryMassachusetts Institute of TechnologyCambridgeUSA
  5. 5.Department of Computer ScienceUniversity of FreiburgFreiburgGermany

Personalised recommendations