1. 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

2. 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

3. (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 »
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Computer Science Department, University of Osaka

4. Self-Stabilization [Dijkstra, 1974]
Example: Dijkstra’s Token Ring 2 1 1 1 1 1
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Computer Science Department, University of Osaka

5. Starting from an arbitrary state5 4 1 5 2 5 4 5
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Computer Science Department, University of Osaka

6. Definition: Closure + Convergence
Legitimate States
Illegitimate States
Convergence
States of the System
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Computer Science Department, University of Osaka

7. 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
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Computer Science Department, University of Osaka

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

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

17. 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
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Computer Science Department, University of Osaka

18. 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

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

20. Computer Science Department, University of Osaka
Algorithm
Assumption: k>∆
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Computer Science Department, University of Osaka

21. 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

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

23. 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

24. Computer Science Department, University of Osaka
Convergence (3/4) 3
(3)
For k=3
1/2
(1/2)2=1/4
1/2
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Computer Science Department, University of Osaka

25. Computer Science Department, University of Osaka
Convergence (4/4) 4
(4)
Trivial from Lemma 3
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Computer Science Department, University of Osaka

26. 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

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

28. 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

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

30. 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

31. 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

32. 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

33. 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

34. Computer Science Department, University of Osaka
The Algorithm
Example N=6 (mN=4) 2 1 1 3
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Computer Science Department, University of Osaka

35. Computer Science Department, University of Osaka
Closure 2 1 1 2 3
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Computer Science Department, University of Osaka

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

37. 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
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Computer Science Department, University of Osaka

38. 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

39. Impossibility for Leader Election (under a distributed scheduler)
Synchronous
Synchronous Execution
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Computer Science Department, University of Osaka

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

41. 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

42. 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

43. 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

44. 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

45. まいど　おおきに !
Maido ookini