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On Stabilization in Herman’s Algorithm

  • Stefan Kiefer
  • Andrzej S. Murawski
  • Joël Ouaknine
  • James Worrell
  • Lijun Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6756)

Abstract

Herman’s algorithm is a synchronous randomized protocol for achieving self-stabilization in a token ring consisting of N processes. The interaction of tokens makes the dynamics of the protocol very difficult to analyze. In this paper we study the expected time to stabilization in terms of the initial configuration.

It is straightforward that the algorithm achieves stabilization almost surely from any initial configuration, and it is known that the worst-case expected time to stabilization (with respect to the initial configuration) is Θ(N 2). Our first contribution is to give an upper bound of 0.64N 2 on the expected stabilization time, improving on previous upper bounds and reducing the gap with the best existing lower bound. We also introduce an asynchronous version of the protocol, showing a similar O(N 2) convergence bound in this case.

Assuming that errors arise from the corruption of some number k of bits, where k is fixed independently of the size of the ring, we show that the expected time to stabilization is O(N). This reveals a hitherto unknown and highly desirable property of Herman’s algorithm: it recovers quickly from bounded errors. We also show that if the initial configuration arises by resetting each bit independently and uniformly at random, then stabilization is significantly faster than in the worst case.

Keywords

Companion Pairing Interact Particle System Synchronous Case Token Ring Neighboring Token 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Stefan Kiefer
    • 1
  • Andrzej S. Murawski
    • 2
  • Joël Ouaknine
    • 1
  • James Worrell
    • 1
  • Lijun Zhang
    • 3
  1. 1.Department of Computer ScienceUniversity of OxfordUK
  2. 2.Department of Computer ScienceUniversity of LeicesterUK
  3. 3.DTU InformaticsTechnical University of DenmarkDenmark

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