Weak vs Self vs Probabilistic Stabilization Stphane Devismes

Weak vs. Self vs. Probabilistic Stabilization Stéphane Devismes (CNRS, LRI, France) Sébastien Tixeuil (LIP 6 -CNRS & INRIA, France) Masafumi Yamashita (Kyushu University, Japan) 19/06/2008 ICDCS'08, Beijing, China

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 » • 19/06/2008 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 ICDCS'08, Beijing, China 1

Definition: Closure + Convergence Closure Illegitimate states Legitimate States Convergence States of the system 19/06/2008 ICDCS'08, Beijing, China 2

Execution of a Self-Stabilizing Algorithm S 19/06/2008 P S … ICDCS'08, Beijing, China 3

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 19/06/2008 ICDCS'08, Beijing, China 4

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 19/06/2008 ICDCS'08, Beijing, China 5

Weak-Stabilization S 19/06/2008 P S … ICDCS'08, Beijing, China 6

Probabilistic Stabilization S 19/06/2008 P S … The expected time before reaching a green segment is finite ICDCS'08, Beijing, China 7

Problem centric point of view • Probabilistic Stabilization • Pseudo-Stabilization • K-Stabilization > Self-stabilization E. g. graph coloring, token passing, alternating bit, … • Open question: Weak-Stabilization 19/06/2008 ICDCS'08, Beijing, China 8

Our Results 1. From a problem centric point of view, Weak- Stabilization > Self-Stabilization 2. Weak-Stabilization & Probabilistic Stabilization are strongly connected 19/06/2008 ICDCS'08, Beijing, China 9

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) 19/06/2008 ICDCS'08, Beijing, China 10

Impossibility for Leader Election (under a distributed scheduler) Synchronous Execution 19/06/2008 ICDCS'08, Beijing, China 11

Weak-Stabilizing Leader Election • Using a parent pointer Par Neig { }, 3 cases: (1) (2) (3) 19/06/2008 ICDCS'08, Beijing, China 12

Why Weak is easier than Self ? • Scheduler in Self-Stabilization: adversary • Scheduler in Weak-Stabilization: friend • Synchronous scheduler: Weak = Self 19/06/2008 ICDCS'08, Beijing, China 13

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) 19/06/2008 ICDCS'08, Beijing, China 14

Problem: Synchronous Case Weak-Stalibiling under a distributed scheduler Random Schedule Synchronous (Asynchronous) Probabilistically Stabilizing In any case Not Probabilistically Stabilizing in the general case Solution: When activated, tosse a coin before moving 19/06/2008 ICDCS'08, Beijing, China 15

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 19/06/2008 ICDCS'08, Beijing, China 16

Thank you 19/06/2008 ICDCS'08, Beijing, China