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

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weak vs self vs probabilistic stabilization

Weak vs. Self vs. Probabilistic Stabilization

StéphaneDevismes(CNRS, LRI, France)

Sébastien Tixeuil (LIP6-CNRS & INRIA, France)

Masafumi Yamashita (Kyushu University, Japan)

introduction
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

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definition closure convergence
Definition: Closure + Convergence

Closure

Legitimate States

Illegitimate states

Convergence

States of the system

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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 stabilization
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

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weaker properties
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

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weak stabilization
Weak-Stabilization

S

P

S

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probabilistic stabilization

S

P

S

Probabilistic Stabilization

The expected time before reaching a green segment is finite

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problem centric point of view
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, …

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our results
Our Results
  • From a problem centric point of view, Weak-Stabilization > Self-Stabilization
  • 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 election

(1)

(2)

(3)

Weak-Stabilizing Leader Election
  • Using a parent pointer Par  Neig  {}, 3 cases:

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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 probabilistic
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)

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problem synchronous case
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

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conclusion
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