第１部: 自己安定の緩和 すてふぁん どぅゔぃむ ポスドク パリ第１１大学 LRI CNRS あどばいざ: せばすちゃ てぃくそい 自己安定　よもやまばなし 第１部:　自己安定の緩和 すてふぁん　どぅゔぃむ ポスドク　パリ第１１大学 LRI CNRS あどばいざ: せばすちゃ　てぃくそい

Computer Science Department, University of Osaka Roadmap Self-Stabilization Advantages Drawbacks Weakened Forms Probabilistic Self-Stabilization Weak-Stabilization 29/07/2008 Computer Science Department, University of Osaka

(Deterministic) Self-Stabilization [Dijkstra, 1974]: « 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 » 29/07/2008 Computer Science Department, University of Osaka

Self-Stabilization [Dijkstra, 1974] Example: Dijkstra’s Token Ring 2 1 1 1 1 1 29/07/2008 Computer Science Department, University of Osaka

Starting from an arbitrary state 5 4 1 5 2 5 4 5 29/07/2008 Computer Science Department, University of Osaka

Definition: Closure + Convergence Legitimate States Illegitimate States Convergence States of the System 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Why is it useful? After a finite period of perturbation, e.g., after an attack hits the system… 5 4 1 5 2 4 5 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Advantages Tolerance to any transient fault: No hypothesis on the nature of extent of transient faults Recovers from the effects of those faults in a unified manner No initialization: Large scale systems Dynamicity: Self-organization in sensor and adhoc networks 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Drawbacks Eventual safety E.g., Mutual Exclusion: a process can execute the CS an unbounded number of times while violating the safety Overhead Self-stabilizing protocols can make use of a large amount of resources Impossibility results Some fundamental problems have no self-stabilizing solution 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Drawbacks To circumvent such drawbacks: Weaker properties (First lecture) Stronger properties (Second lecture) 29/07/2008 Computer Science Department, University of Osaka

Weakened Forms of Self-Stabilization Pseudo-Stabilization K-Stabilization Probabilistic Stabilization Weak-Stabilization Aim: Circumvent impossibility results E.g., Coloring, Leader Election, Token Circulation 29/07/2008 Computer Science Department, University of Osaka

Pseudo-Stabilization [Burns et al, WSS’89] « Starting from any configuration, any execution converges to a correct suffix » Self-Stabilization: « Starting from any configuration, any execution converges to a configuration from which any suffix is correct » Same convergence + weakened closure 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Pseudo- vs. Self- … S P S Finite Number Unbounded Time Consequence: the convergence time to the correct suffix is unbounded in the general case 29/07/2008 Computer Science Department, University of Osaka

K-Stabilization [Beauquier et al, PODC’98] Self-Stabilization: Any configuration can be initial K-Stabilization: Only a subset of configurations can be initial: any configuration that can be obtained with at most K faults 29/07/2008 Computer Science Department, University of Osaka

Probabilistic Self-Stabilization [Israeli and Jalfon, PODC’90] Same closure But a weakened convergence: The execution converges with probability 1 The expected convergence time is finite 29/07/2008 Computer Science Department, University of Osaka

Probabilistic Stabilization … The expected time before reaching a green segment is finite 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Case Study: Coloring in Unidirectional (General) Network [Under submission in SSS’08] Joined work with Samuel Bernard, Maria Gradinariu, and Sébastien Tixeuil 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Impossibility result No deterministic self-stabilizing solution under a distributed scheduler 29/07/2008 Computer Science Department, University of Osaka

Remark: More Difficult than the bidirectional case 29/07/2008 In the unidirectional case, the number of « conflicts » can increase! Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Algorithm Assumption: k>∆ 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Proof Closure: Trivial Once there is a coloring, no process moves anymore 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Convergence (1/4) Example: Ring, 1/2 for k=3 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Convergence (2/4) 2 (2) Maximization of Lemma 1: Remark: for the ring, M = 1/2 for k = 3 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Convergence (3/4) 3 (3) For k=3 1/2 (1/2)2=1/4 1/2 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Convergence (4/4) 4 (4) Trivial from Lemma 3 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Conclusion With the minimal number of color (∆+1), we obtain an expected convergence time of at most n.∆ steps When k is huge (k), the expected convergence time is near n steps 29/07/2008 Computer Science Department, University of Osaka

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

Weak-Stabilization [Gouda, WSS’01] Same closure property But weakened convergence: From any configuration, there is at least one possible execution which converges 29/07/2008 Computer Science Department, University of Osaka

Weak-Stabilization [Gouda, WSS’01] … S P S 29/07/2008 Computer Science Department, University of Osaka

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, … 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Our Results From a problem centric point of view, Weak-Stabilization > Self-Stabilization Weak-Stabilization & Probabilistic Stabilization are strongly connected 29/07/2008 Computer Science Department, University of Osaka

Weak > Self (Problem centric point of view) Two examples: Token Circulation in unidirectional anonymous rings under a distributed scheduler Leader Election in anonymous tree under a distributed scheduler (2 algorithms) 29/07/2008 Computer Science Department, University of Osaka

Token Circulation in Unidirectional Anonymous Rings [Herman, IPL’90]: « no deterministic self-stabilizing solution » We prove that the K-fair algorithm of [Beauquier et al, DC’07] is actually a weak-stabilizing token circulation 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka The Algorithm Example N=6 (mN=4) 2 1 1 3 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Closure 2 1 1 2 3 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Convergence (1/2) There always exists at least one token 3 3 2 1 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka Convergence (2/2) If the number of tokens is greater than 1, it is possible to make it decrease 3 2 1 1 3 29/07/2008 Computer Science Department, University of Osaka

Leader Election in Anonymous Tree under a Distributed Scheduler We prove that there is no deterministic self-stabilizing solution We give two examples of weak-stabilizing leader election 29/07/2008 Computer Science Department, University of Osaka

Impossibility for Leader Election (under a distributed scheduler) Synchronous Synchronous Execution 29/07/2008 Computer Science Department, University of Osaka

Weak-Stabilizing Leader Election Using a parent pointer Par  Neig  {}, 3 cases: (1) (2) (3) 29/07/2008 Computer Science Department, University of Osaka

Why Weak is easier than Self ? Scheduler in Self-Stabilization: Adversary Scheduler in Weak-Stabilization: Friend Synchronous Scheduler: Weak = Self 29/07/2008 Computer Science Department, University of Osaka

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) Dessin ?? Patate + arbres 29/07/2008 Computer Science Department, University of Osaka

Problem: Synchronous Case Weak-Stabilizing under a distributed scheduler Random Schedule (Asynchronous) Synchronous Probabilistically Stabilizing in any case Not Probabilistically Stabilizing in the general case Solution: When activated, toss a coin before moving 29/07/2008 Computer Science Department, University of Osaka

Computer Science Department, University of Osaka 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 29/07/2008 Computer Science Department, University of Osaka

まいど　おおきに ! Maido ookini