Full Reversal Routing as a Linear Dynamical System

  • Bernadette Charron-Bost
  • Matthias Függer
  • Jennifer L. Welch
  • Josef Widder
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6796)

Abstract

Link reversal is a versatile algorithm design paradigm, originally proposed by Gafni and Bertsekas in 1981 for routing, and subsequently applied to other problems including mutual exclusion and resource allocation. Although these algorithms are well-known, until now there have been only preliminary results on time complexity, even for the simplest link reversal scheme for routing, called Full Reversal (FR). In this paper we tackle this open question for arbitrary communication graphs. Our central technical insight is to describe the behavior of FR as a dynamical system, and to observe that this system is linear in the min-plus algebra. From this characterization, we derive the first exact formula for the time complexity: Given any node in any (acyclic) graph, we present an exact formula for the time complexity of that node, in terms of some simple properties of the graph. These results for FR are instrumental in analyzing a broader class of link reversal routing algorithms, as we show in a companion paper that such algorithms can be reduced to FR. In the current paper, we further demonstrate the utility of our formulas by using them to show the previously unknown fact that FR is time-efficient when executed on trees.

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References

  1. 1.
    Gafni, E., Bertsekas, D.P.: Distributed algorithms for generating loop-free routes in networks with frequently changing topology. IEEE Transactions on Communications 29, 11–18 (1981)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Chandy, K.M., Misra, J.: The drinking philosopher’s problem. ACM Transactions on Programming Languages and Systems 6(4), 632–646 (1984)CrossRefGoogle Scholar
  3. 3.
    Barbosa, V.C., Gafni, E.: Concurrency in heavily loaded neighborhood-constrained systems. ACM Trans. Program. Lang. Syst. 11(4), 562–584 (1989)CrossRefGoogle Scholar
  4. 4.
    Malka, Y., Moran, S., Zaks, S.: A lower bound on the period length of a distributed scheduler. Algorithmica 10(5), 383–398 (1993)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Tirthapura, S., Herlihy, M.: Self-stabilizing distributed queuing. IEEE Transactions on Parallel and Distributed Systems 17(7), 646–655 (2006)CrossRefMATHGoogle Scholar
  6. 6.
    Attiya, H., Gramoli, V., Milani, A.: A provably starvation-free distributed directory protocol. In: 12th International Symposium on Stabilization, Safety, and Security of Distributed Systems, pp. 405–419 (2010)Google Scholar
  7. 7.
    Park, V.D., Corson, M.S.: A highly adaptive distributed routing algorithm for mobile wireless networks. In: 16th Conference on Computer Communications (Infocom), apr 1997, pp. 1405–1413 (1997)Google Scholar
  8. 8.
    Ko, Y.-B., Vaidya, N.H.: Geotora: a protocol for geocasting in mobile ad hoc networks. In: Proceedings of the 2000 International Conference on Network Protocols, ICNP 2000, pp. 240–250 (2000)Google Scholar
  9. 9.
    Raymond, K.: A tree-based algorithm for distributed mutual exclusion. ACM Transactions on Computer Systems 7(1), 61–77 (1989)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Naimi, M., Trehel, M., Arnold, A.: A log(n) distributed mutual exclusion algorithm based on path reversal. Journal on Parallel and Distributed Computing 34(1), 1–13 (1996)CrossRefGoogle Scholar
  11. 11.
    Walter, J.E., Welch, J.L., Vaidya, N.H.: A mutual exclusion algorithm for ad hoc mobile networks. Wireless Networks 7(6), 585–600 (2001)CrossRefMATHGoogle Scholar
  12. 12.
    L., J., Malpani, N.V.N., Welch: Leader election algorithms for mobile ad hoc networks. In: Proceedings of the 4th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communication (2000)Google Scholar
  13. 13.
    Derhab, A., Badache, N.: A self-stabilizing leader election algorithm in highly dynamic ad hoc mobile networks. IEEE Trans. Parallel Distrib. Syst. 19(7), 926–939 (2008)CrossRefMATHGoogle Scholar
  14. 14.
    Ingram, R., Shields, P., Walter, J.E., Welch, J.L.: An asynchronous leader election algorithm for dynamic networks. In: Proceedings of the IEEE International Parallel & Distributed Processing Symposium, pp. 1–12 (2009)Google Scholar
  15. 15.
    Busch, C., Surapaneni, S., Tirthapura, S.: Analysis of link reversal routing algorithms for mobile ad hoc networks. In: Proceedings of the 15th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 210–219 (2003)Google Scholar
  16. 16.
    Busch, C., Tirthapura, S.: Analysis of link reversal routing algorithms. SIAM Journal on Computing 35(2), 305–326 (2005)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Charron-Bost, B., Gaillard, A., Welch, J.L., Widder, J.: Routing without ordering. In: Proceedings of the 21st ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 145–153 (2009)Google Scholar
  18. 18.
    Charron-Bost, B., Welch, J.L., Widder, J.: Link reversal: How to play better to work less. In: Dolev, S. (ed.) ALGOSENSORS 2009. LNCS, vol. 5804, pp. 88–101. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Charron-Bost, B., Függer, M., Welch, J.L., Widder, J.: Partial is full. In: Kosowski, A., Yamashita, M. (eds.) SIROCCO 2011. LNCS, vol. 6796, pp. 111–123. Springer, Heidelberg (2011)Google Scholar
  20. 20.
    Charron-Bost, B., Függer, M., Welch, J.L., Widder, J.: Full reversal routing as a linear dynamical system. Research Report 7/2011, Technische Universität Wien, Institut für Technische Informatik, Treitlstr. 1-3/182-2, 1040 Vienna, Austria (2011)Google Scholar
  21. 21.
    Heidergott, B., Olsder, G.J., von der Woude, J.: Max plus at work. Princeton Univ. Press, Princeton (2006)Google Scholar
  22. 22.
    Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.-P.: Synchronization and Linearity. John Wiley & Sons, Chichester (1993)MATHGoogle Scholar
  23. 23.
    Malka, Y., Rajsbaum, S.: Analysis of distributed algorithms based on recurrence relations (preliminary version). In: Toueg, S., Kirousis, L.M., Spirakis, P.G. (eds.) WDAG 1991. LNCS, vol. 579, pp. 242–253. Springer, Heidelberg (1992)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Bernadette Charron-Bost
    • 1
  • Matthias Függer
    • 2
  • Jennifer L. Welch
    • 3
  • Josef Widder
    • 3
  1. 1.CNRS, LIXEcole polytechniquePalaiseauFrance
  2. 2.TU WienAustria
  3. 3.Texas A&M UniversityUSA

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