Weak vs. Self vs. Probabilistic Stabilization

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

2. 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: • Tolerance to any transient fault • No hypothesis on the nature or extent of faults • Recovers from the effects of those faults in a unified manner ICDCS'08, Beijing, China

3. Definition: Closure + Convergence Closure Legitimate States Illegitimate states Convergence States of the system ICDCS'08, Beijing, China

4. Execution of a Self-Stabilizing Algorithm … S P S ICDCS'08, Beijing, China

5. Drawbacks of Self-Stabilization • May make use of a large amount of resources • May be difficult to design and to prove • Could be unable to solve some fundamental problems ICDCS'08, Beijing, China

6. 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 ICDCS'08, Beijing, China

7. Weak-Stabilization … S P S ICDCS'08, Beijing, China

8. … S P S Probabilistic Stabilization The expected time before reaching a green segment is finite ICDCS'08, Beijing, China

9. 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, … ICDCS'08, Beijing, China

10. Our Results • From a problem centric point of view, Weak-Stabilization > Self-Stabilization • Weak-Stabilization & Probabilistic Stabilization are strongly connected ICDCS'08, Beijing, China

11. 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) ICDCS'08, Beijing, China

12. Impossibility for Leader Election(under a distributed scheduler) Synchronous Execution ICDCS'08, Beijing, China

13. (1) (2) (3) Weak-Stabilizing Leader Election • Using a parent pointer Par  Neig  {}, 3 cases: ICDCS'08, Beijing, China

14. Why Weak is easier than Self ? • Scheduler in Self-Stabilization: adversary • Scheduler in Weak-Stabilization: friend • Synchronous scheduler: Weak = Self ICDCS'08, Beijing, China

15. 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 OutlineExecution: random walk in a finite set (of configurations) ICDCS'08, Beijing, China

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

17. 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 ICDCS'08, Beijing, China

18. Thank you