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Randomized self-stabilizing and space optimal leader election under arbitrary scheduler on rings

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Abstract

We present a randomized self-stabilizing leader election protocol and a randomized self-stabilizing token circulation protocol under an arbitrary scheduler on anonymous and unidirectional rings of any size. These protocols are space optimal. We also give a formal and complete proof of these protocols. To this end, we develop a complete model for probabilistic self-stabilizing distributed systems which clearly separates the non deterministic behavior of the scheduler from the randomized behavior of the protocol. This framework includes all the necessary tools for proving the self- stabilization of a randomized distributed system: definition of a probabilistic space and definition of the self-stabilization of a randomized protocol. We also propose a new technique of scheduler management through a self-stabilizing protocol composition (cross-over composition). Roughly speaking, we force all computations to have a fairness property under any scheduler, even under an unfair one.

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References

  1. Anagnostou, E., El-Yaniv, R.: More on the power of random walks—uniform self-stabilizing randomized algorithms. In: WDAG91, Distributed Algorithms 5th International Workshop Proceedings. LNCS, vol. 579, pp. 31–51. Springer, Heidelberg (1991)

  2. Angluin, D.: Local and global properties in networks of processors. In: STOC80, the 12th Annual ACM Symposium on Theory of Computing, pp. 82–93. ACM (1980)

  3. Awerbuch, B., Ostrovsky, R.: Memory-efficient and self-stabilizing network reset. In: PODC94, the 13th Annual ACM Symposium on Principles of Distributed Computing, pp. 254–263 (1994)

  4. Beauquier, J., Cordier, S., Delaët, S.: Optimum probabilistic self-stabilization on uniform rings. In: WSS95, the 2nd Workshop on Self-Stabilizing Systems, pp. 15.1–15.15 (1995)

  5. Beauquier J., Durand-Lose J., Gradinariu M. and Johnen C. (2002). Token based self-stabilizing uniform algorithms. J Parallel Distrib. Comput. 62(5): 899–921

    Article  MATH  Google Scholar 

  6. Beauquier, J., Gradinariu, M., Johnen, C.: Memory space requirements for self-stabilizing leader election protocols. In: PODC99, the 18th Annual ACM Symposium on Principles of Distributed Computing, pp. 199–208 (1999)

  7. Beauquier, J., Gradinariu, M., Johnen, C.: Randomized self- stabilizing and space optimal leader election under arbitrary scheduler on rings. Technical Report 1225, L.R.I, December (1999)

  8. Beauquier, J., Gradinariu, M., Johnen, C.: Cross-over composition - enforcement of fairness under unfair adversary. In: WSS01, the 5th International Workshop on Self-Stabilizing Systems. LNCS, vol. 2194, pp. 19–34. Springer, Heidelberg (2001)

  9. Beauquier, J., Johnen, C., Messika, S.: All k-bounded policies are equivalent for self-stabilization. In: SSS’06, the 8th International Symposium on Stabilization, Safety, and Security of Distributed Systems. LNCS. Springer, Heidelberg (2006)

  10. Billingsley P. (1986). Probability and Measure. Wiley

  11. Boulinier, C., Petit, F., Villain, V.: When graph theory helps self-stabilization. In: PODC04, the 23th Annual ACM Symposium on Principles of Distributed Computing, pp. 150–160 (2004)

  12. Christoff, I.: Testing equivalences and fully abstract models for probabilistic processes. In: CONCUR90, the 1st International Conference on Concurrency Theory. LNCS, vol. 458, pp. 126–140, Springer, Heidelberg (1990)

  13. Datta, A.K., Tixeuil, S., Gradinariu, M.: Self-stabilizing mutual exclusion under arbitrary scheduler. Comput. J. 47(1) (2004)

  14. de Alfaro, L.: Formal Verification of Probabilistic systems. PhD Thesis, Stanford University, (1997)

  15. Défago, X., Gradinariu, M., Messika, S., Raipin Parvédy, P.: Fault-tolerant and self-stabilizing mobile robots gathering. In: DISC06, the 20th International Conference on Distributed Computing. LNCS, vol. 3274, pp. 46–60. Springer, Heidelberg (2006)

  16. Dolev, S., Gouda, M.G., Schneider, M.: Memory requirements for silent stabilization. In: PODC96, the 15th Annual ACM Symposium on Principles of Distributed Computing, pp. 27–34 (1996)

  17. Dolev, S., Herman, T.: Parallel composition of stabilizing algorithms. In: WSS99, the 4th Workshop on Self-Stabilizing Systems (published in association with ICDCS99 The 19th IEEE International Conference on Distributed Computing Systems), pp. 25–32 (1999)

  18. Dolev S., Israeli A. and Moran S. (1993). Self-stabilization of dynamic systems assuming only Read/Write atomicity. Distrib. Comput. 7: 3–16

    Article  Google Scholar 

  19. Dolev S., Israeli A. and Moran S. (1995). Analyzing expected time by scheduler-luck games. IEEE Trans. Softw. Eng. 21: 429–439

    Article  Google Scholar 

  20. Duchon, P., Hanusse, N., Tixeuil, S.: Optimal randomized self- stabilizing mutual exclusion on synchronous rings. In: DISC04, the 18th International Conference on Distributed Computing. LNCS, vol. 3274, pp. 216–229 Springer, Heidelberg (2004)

  21. Duflot M., Fribourg L. and Picaronny C. (2004). Randomized dining philosophers without fairness assumption. Distrib. Comput. 17(1): 65–76

    Article  Google Scholar 

  22. Fribourg L., Messika S. and Picaronny C. (2006). Coupling and self- stabilization. Distrib. Comput. 18(3): 221–232

    Article  Google Scholar 

  23. Gouda, M.G., Haddix, F.: The alternator. In: WSS99, the 4th Workshop on Self-Stabilizing Systems (published in association with ICDCS99 The 19th IEEE International Conference on Distributed Computing Systems), pp. 48–53 (1999)

  24. Gouda M.G. and Herman T. (1991). Adaptive programming. IEEE Trans. Softw. Eng. 17: 911–921

    Article  Google Scholar 

  25. Gradinariu, M., Johnen, C.: Self-stabilizing neighborhood unique naming under unfair scheduler. In: Euro-Par’01 Parallel Processing. LNCS, vol. 2150, pp. 458–465, Springer, Heidelberg (2001)

  26. Gradinariu, M., Tixeuil, S.: Self-stabilizing vertex coloration and arbitrary graphs. In: OPODIS’00, 4th International Conference On Principles Of DIstributed Systems, pp. 55–70 (2000)

  27. Israeli, A., Jalfon, M.: Token management schemes and random walks yield self-stabilizing mutual exclusion. In: PODC90, the 9th Annual ACM Symposium on Principles of Distributed Computing, pp. 119–131 (1990)

  28. Itkis, G., Levin, L.: Fast and lean self-stabilizing asynchronous protocols. In: FOCS94, the 34th Annual IEEE Symposium on Foundations of Computer Science, pp. 226–239 (1994)

  29. Johnen, C.: Service time optimal self-stabilizing token circulation protocol on anonymous unidrectional. In: SRDS02, the 21th IEEE Symposium on Reliable Distributed Systems, pp. 80–89. IEEE Computer Society Press, (2002)

  30. Johnen, C.: Bounded service time and memory space optimal self-stabilizing token circulation protocol on unidirectional rings. In: IPDPS’04, the 18th IEEE International Parallel & Distributed Processing Symposium (2004)

  31. Karaata M.H. (2001). Self-stabilizing strong fairness under weak fairness. IEEE Trans. Parallel Distrib. Syst. 12(4): 337–345

    Article  Google Scholar 

  32. Lehmann, D., Rabin, M.O.: On the advantages of free choice: a symmetric and fully-distributed solution to the dining philosophers problem. In: POPL81, the 8th Annual ACM Symposium on Principles of Programming Languages, pp. 133–138 (1981)

  33. Mayer, A., Ofek, Y., Ostrovsky, R., Yung, M.: Self-stabilizing symmetry breaking in constant-space. In: STOC92, the 24th Annual ACM Symposium on Theory of Computing, pp. 667–678 (1992)

  34. Nesterenko M. and Arora A. (2002). Stabilization-preserving atomicity refinement. J. Parallel Distrib. Comput. 62(5): 766–791

    Article  MATH  Google Scholar 

  35. Pnueli A. and Zuck L. (1986). Verification of multiprocess probabilistic protocols. Distrib. Comput. 1(1): 53–72

    Article  MATH  Google Scholar 

  36. Pogosyants, A., Segala, R.: Formal verification of timed properties of randomized distributed algorithms. In: PODC95, the 14th Annual ACM Symposium on Principles of Distributed Computing, pp. 174–183 (1995)

  37. Rabin, M.O.: Randomized byzantine generals. In: FOCS84, the 24st Annual IEEE Symposium on Foundations of Computer Science, pp. 403–409 (1984)

  38. Rosaz, L.: Self-stabilizing token circulation on asynchronous uniform unidirectional rings. In: PODC00, the 19th Annual ACM Symposium on Principles of Distributed Computing, pp. 249–258 (2000)

  39. Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, MIT, Departament of Electrical Engineering and Computer Science (1995)

  40. Segala, R., Lynch, N.: Probabilistic simulations for probabilistic processes. In: CONCUR94, the 5th International Conference on Concurrency Theory. LNCS, vol. 836, pp. 481–496. Springer, Heidelberg (1994)

  41. van Glabbeek, R.J., Smolka, S.A., Steffen, B., Toft, C.M.N.: Reactive, generative and stratified models of probabilistic processes. In: LICS90, the 5th Annual IEEE Symposium on Logic in Computer Science (1990)

  42. Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state programs. In: FOCS85, the 25st Annual IEEE Symposium on Foundations of Computer Science, pp. 327–338, (1985)

  43. Varghese, G.: Compositional proofs of self-stabilizing protocols. In WSS97, the 3rd Workshop on Self-Stabilizing Systems, pp. 80–94. Carleton University Press (1997)

  44. Wu, S.H., Smolka, S.A., Stark, E.W.: Composition and behaviors of probabilistic I/O automata. In: CONCUR94, the 5th International Conference on Concurrency Theory. LNCS, vol. 836, pp. 513–528 Springer, Heidelberg (1994)

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Authors and Affiliations

  1. LRI, Univ. Paris-Sud, CNRS, Bat 490, 91405, Orsay, France

    Joffroy Beauquier & Colette Johnen

  2. LIP6, Univ. Paris 6, CNRS, INRIA, Paris, France

    Maria Gradinariu

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  1. Joffroy Beauquier
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  2. Maria Gradinariu
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  3. Colette Johnen
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Correspondence to Colette Johnen.

Additional information

This work was done while Maria Gradinariu was working at LRI, Univ. Paris-Sud, CNRS.

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Beauquier, J., Gradinariu, M. & Johnen, C. Randomized self-stabilizing and space optimal leader election under arbitrary scheduler on rings. Distrib. Comput. 20, 75–93 (2007). https://doi.org/10.1007/s00446-007-0034-0

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  • DOI: https://doi.org/10.1007/s00446-007-0034-0

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