Weak vs. Self vs. Probabilistic Stabilization Stéphane Devismes (CNRS, LRI, France) Sébastien Tixeuil (LIP6-CNRS & INRIA, France) Masafumi Yamashita (Kyushu University, Japan)

19/06/2008ICDCS'08, Beijing, China1 Introduction (Deterministic) Self-Stabilization:  « A protocol P is self-stabilizing if, starting from any initial configuration, every execution of P eventually reaches a point from which its behaviour is correct » Advantages: 1.Tolerance to any transient fault 2.No hypothesis on the nature or extent of faults 3.Recovers from the effects of those faults in a unified manner

19/06/2008ICDCS'08, Beijing, China2 Definition: Closure + Convergence States of the system Illegitimate states Legitimate States Convergence Closure

19/06/2008ICDCS'08, Beijing, China3 Execution of a Self-Stabilizing Algorithm SPS …

19/06/2008ICDCS'08, Beijing, China4 Drawbacks of Self-Stabilization 1. May make use of a large amount of resources 2. May be difficult to design and to prove 3. Could be unable to solve some fundamental problems

19/06/2008ICDCS'08, Beijing, China5 Weaker Properties Probabilistic Stabilization [Israeli and Jalfon, PODC’90]: probabilistic convergence Pseudo stabilization [Burns et al, WSS’89]: always a correct suffix K-stabilization [Beauquier et al, PODC’98]: at most K faults in the initial configuration Weak-Stabilization [Gouda, WSS’01]: from any configuration, there is at least one possible execution which converges

19/06/2008ICDCS'08, Beijing, China6 Weak-Stabilization SPS …

19/06/2008ICDCS'08, Beijing, China7 Probabilistic Stabilization SPS … The expected time before reaching a green segment is finite

19/06/2008ICDCS'08, Beijing, China8 Problem centric point of view Probabilistic Stabilization Pseudo-Stabilization K-Stabilization Open question: Weak-Stabilization > Self-stabilization E.g. graph coloring, token passing, alternating bit, …

19/06/2008ICDCS'08, Beijing, China9 Our Results 1. From a problem centric point of view, Weak-Stabilization > Self-Stabilization 2. Weak-Stabilization & Probabilistic Stabilization are strongly connected

19/06/2008ICDCS'08, Beijing, China10 Weak > Self (Problem centric point of view) Two examples:  Token Circulation in unidirectional rings under a distributed scheduler  Leader Election in anonymous tree under a distributed scheduler (2 algorithms)

19/06/2008ICDCS'08, Beijing, China11 Impossibility for Leader Election (under a distributed scheduler) Synchronous Execution

19/06/2008ICDCS'08, Beijing, China12 Weak-Stabilizing Leader Election Using a parent pointer Par  Neig  {  }, 3 cases: (1) (2) (3)

19/06/2008ICDCS'08, Beijing, China13 Why Weak is easier than Self ? Scheduler in Self-Stabilization: adversary Scheduler in Weak-Stabilization: friend Synchronous scheduler: Weak = Self

19/06/2008ICDCS'08, Beijing, China14 Observation: Weak vs. Probabilistic If a protocol P has a finite number of configurations, then P is weak-stabilizing iff P is probabilistically stabilizing under a randomized scheduler Outline Execution: random walk in a finite set (of configurations)

19/06/2008ICDCS'08, Beijing, China15 Problem: Synchronous Case Weak-Stalibiling under a distributed scheduler Probabilistically Stabilizing In any case Not Probabilistically Stabilizing in the general case Random Schedule (Asynchronous) Synchronous Solution: When activated, tosse a coin before moving

19/06/2008ICDCS'08, Beijing, China16 Conclusion From the problem centric point of view, Weak-Stabilization > Self-Stabilization Weak-Stabilization = Probabilistic Stabilization if the scheduler is probabilistic and the set of configurations is finite  Interesting in practical settings: Weak-Stabilization is easier to design than probabilistic stabilization In real systems, the scheduler behaves randomly Can be easily transformed to support the synchronous scheduler Perspective: evaluating a expected convergence time

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