Weak vs. Self vs. Probabilistic Stabilization

Weak vs. Self vs. Probabilistic Stabilization

Stéphane Devismes (CNRS, LRI, France)

Sébastien Tixeuil (LIP6-CNRS & INRIA, France)

Masafumi Yamashita (Kyushu University, Japan)

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IntroductionIntroduction

• (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

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Definition: Closure + ConvergenceDefinition: Closure + Convergence

States of the system

Illegitimate states Legitimate States

Convergence

Closure

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Execution of a Self-Stabilizing Algorithm

Execution of a Self-Stabilizing Algorithm

S P S …

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Drawbacks of Self-StabilizationDrawbacks 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

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Weaker PropertiesWeaker 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

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Weak-StabilizationWeak-Stabilization

S P S …

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Probabilistic StabilizationProbabilistic Stabilization

S P S … The expected time before reaching a green segment is finite

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Problem centric point of viewProblem centric point of view

• Probabilistic Stabilization

• Pseudo-Stabilization

• K-Stabilization

• Open question: Weak-Stabilization

> Self-stabilizationE.g. graph coloring, token passing, alternating bit, …

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Our ResultsOur Results

1. From a problem centric point of view,

Weak-Stabilization > Self-Stabilization

2. Weak-Stabilization & Probabilistic Stabilization

are strongly connected

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Weak > Self (Problem centric point of view)

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)

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Impossibility for Leader Election(under a distributed scheduler)

Impossibility for Leader Election(under a distributed scheduler)

Synchronous Execution

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Weak-Stabilizing Leader ElectionWeak-Stabilizing Leader Election

• Using a parent pointer Par Neig {}, 3 cases:

(1)

(2)

(3)

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Why Weak is easier than Self ?Why Weak is easier than Self ?

• Scheduler in Self-Stabilization: adversary

• Scheduler in Weak-Stabilization: friend

• Synchronous scheduler: Weak = Self

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Observation: Weak vs. ProbabilisticObservation: 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)

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Problem: Synchronous Case 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

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ConclusionConclusion

• 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

Thank youThank you