1. New Variants of Self-Stabilization
Stéphane Devismes

2. Retraite REGAL 2017, Cap Hornu
Roadmap
Probabilistic Snap-Stabilization
[Altisen & Devismes, TCS, 2017]
Gradual Stabilization
[Altisen et al, EUROPAR’2016]
Retraite REGAL 2017, Cap Hornu

3. Retraite REGAL 2017, Cap Hornu
Roadmap
Probabilistic Snap-Stabilization
[Altisen & Devismes, TCS, 2017]
Gradual Stabilization
[Altisen et al, EUROPAR’2016]
Retraite REGAL 2017, Cap Hornu

4. Self-Stabilization [Dijkstra,74]
Retraite REGAL 2017, Cap Hornu

5. Self-Stabilization [Dijkstra,74]
Transient faults:
Location: node or link
Duration: finite
Frequency: low
e.g., memory corruptions, message losses, message corruptions
Retraite REGAL 2017, Cap Hornu

6. Self-Stabilization [Dijkstra,74]
Retraite REGAL 2017, Cap Hornu

7. Self-Stabilization [Dijkstra,74]
Retraite REGAL 2017, Cap Hornu

8. Self-Stabilization [Dijkstra,74]
Retraite REGAL 2017, Cap Hornu

9. Self-Stabilization [Dijkstra,74]
Retraite REGAL 2017, Cap Hornu

10. Self-Stabilization [Dijkstra,74]
Retraite REGAL 2017, Cap Hornu

11. Self-Stabilization [Dijkstra,74]
Recover
after any number of transient faults
Retraite REGAL 2017, Cap Hornu

12. Retraite REGAL 2017, Cap Hornu
Advantages
Tolerate any finite number of transient faults
Lightweight
Low overhead
No initialization
Self-organization in wireless sensor network
Tolerate (detectable) topological changes
Self-stabilizing algorithms can be easily composed
Retraite REGAL 2017, Cap Hornu

13. Drawbacks of Self-Stabilization
Temporary Loss of Safety
No local detection of stabilization
Permanent local checks
Endlessly repeat computations
Impossibility Results
Retraite REGAL 2017, Cap Hornu

14. Drawbacks of Self-Stabilization
Temporary Loss of Safety
No local detection of stabilization
Permanent local checks
Endlessly repeat computations
Impossibility Results
Retraite REGAL 2017, Cap Hornu

15. Temporary Loss of Safety
Goal: Minimize the stabilization time
Retraite REGAL 2017, Cap Hornu

16. Temporary Loss of Safety
Stabilization time usually in Ω(D) rounds
[Tixeuil & Genolini, 2002]
Retraite REGAL 2017, Cap Hornu

17. Self-Stabilization: endless repetitions of finite task executions
Transient faults
Retraite REGAL 2017, Cap Hornu

18. Self-Stabilization: endless repetitions of finite task executions
Transient faults
Finite (generally unbounded) number of times
Retraite REGAL 2017, Cap Hornu

19. (Deterministic) Snap-Stabilization [Bui et al, 1999]
Resume a correct behavior since the first started task after the end of faults
Transient faults
First start
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20. (Deterministic) Snap-Stabilization [Bui et al, 1999]
Strengthened form of self-stabilization
Snap-Stabilization ⇒ Self-Stabilization
Lot of solutions are available, but only in
Identified, or
Rooted networks
What about anonymous networks ?
Retraite REGAL 2017, Cap Hornu

21. (Deterministic) Snap-Stabilization [Bui et al, 1999]
Most of classical problems even have no deterministic self-stabilizing solutions, e.g.,
Token passing
Leader Election
Several weakened forms of self-stabilization have been introduced to circumvent these impossibility results, e.g.,
Weak-stabilization [Gouda, 2001]
K-stabilization [Beauquier et al, 1998]
Probabilistic self-stabilization [Herman, 1990]
Retraite REGAL 2017, Cap Hornu

22. (Deterministic) Snap-Stabilization [Bui et al, 1999]
Most of classical problems even have no deterministic self-stabilizing solutions, e.g.,
Token passing
Leader Election
Several weakened forms of self-stabilization have been introduced to circumvent these impossibility results, e.g.,
Weak-stabilization [Gouda, 2001]
K-stabilization [Beauquier et al, 1998]
Probabilistic self-stabilization [Herman, 1990]
Retraite REGAL 2017, Cap Hornu

23. Deterministic vs. Probabilistic Self-Stabilization
Deterministic Self-stabilization
Deterministic convergence:
Starting from any configuration, the system converges to a legitimate configuration within finite time
Probabilistic Self-stabilization
Probabilistic convergence:
Starting from any configuration, the system converges to a legitimate configuration within almost surely finite time (Las Vegas Approach)
Retraite REGAL 2017, Cap Hornu

24. Retraite REGAL 2017, Cap Hornu
Contribution (1/2)
Probabilistic Snap-stabilization
Weakened form of snap-stabilization
Not comparable to self-stabilization
Ideas
We relax the definition of snap-stabilization without altering its strong safety guarantees
to address anonymous networks
Retraite REGAL 2017, Cap Hornu

25. Definition For every task started after the faults
(i.e., starting from an arbitrary configuration)
The safety part of the specification is satisfied
However, the liveness part of the specification is only almost surely ensured (Las Vegas approach)
Transient faults
The task is executed safely, but it terminates in almost surely finite time
Retraite REGAL 2017, Cap Hornu

26. Retraite REGAL 2017, Cap Hornu
Contribution (2/2)
2 probabilistic snap-stabilizing protocols for
the guaranteed service leader election
in anonymous networks
using the locally shared memory model where each process knows a value B s.t. B < n ≤ 2B
The first one assumes a synchronous scheduler
The second one assumes an asynchronous scheduler
This problem has no deterministic self- or snap- stabilizing solution
Retraite REGAL 2017, Cap Hornu

27. Definition of the Problem
processes P1 P2 P3 P4 P5
time
Retraite REGAL 2017, Cap Hornu

28. Definition of the Problem
processes ? P1 ? P2 ? P3 ? P4
Upon a request
A process initiates the question
“Am I the leader of the network ?” ? P5
time
Retraite REGAL 2017, Cap Hornu

29. Definition of the problem
The task consists in computing the answer (yes/no)
processes ?
yes P1 ? no P2 ? no P3 ? no P4 ? no P5
time
Retraite REGAL 2017, Cap Hornu

30. Definition of the problem
Each process always receives the same answer to each of its request
processes ? ? ?
yes
yes
yes P1 ? no ? no P2 ? no ? no P3 ? no ? no P4 ? no ? no ? no P5
time
Retraite REGAL 2017, Cap Hornu

31. Definition of the problem
Exactly one process always receives yes
processes ? ? ?
yes
yes
yes P1 ? no ? no P2 ? no ? no P3 ? no ? no P4 ? no ? no ? no P5
time
Retraite REGAL 2017, Cap Hornu

32. Retraite REGAL 2017, Cap Hornu
Complexity
We proposed 2 probabilistic snap-stabilizing protocols for the guaranteed service leader election in anonymous networks where each process knows a value B s.t. B < n ≤ 2B
The first one assumes a synchronous scheduler, and its expected time complexity is in O(n)
The second one assumes an asynchronous scheduler, and its expected time complexity is in O(n2)
These two expected times can be reduced to O(D) and O(Dn), respectively, if processes have the knowledge of the diameter D of the network.
Retraite REGAL 2017, Cap Hornu

33. Retraite REGAL 2017, Cap Hornu
Generalization
Guaranteed service leader election → guaranteed service naming:
we can mimic the behavior of an identified network and emulate the transformer proposed in [Cournier et al, TCS’2016]
CCL: In the locally shared memory model, almost all (non-stabilizing) algorithm  can be, semi-automatically, turned into probabilistic snap-stabilizing algorithm working in anonymous networks.
Perspective: The transformer is heavy!
Need of dedicated solutions with reasonable complexities
Retraite REGAL 2017, Cap Hornu

34. Retraite REGAL 2017, Cap Hornu
Roadmap
Probabilistic Snap-Stabilization
[Altisen & Devismes, TCS, 2017]
Gradual Stabilization
[Altisen et al, EUROPAR’2016]
Retraite REGAL 2017, Cap Hornu

35. Retraite REGAL 2017, Cap Hornu
Related Work
Self-Stab + Safe Convergence
[Kakugawa & Masuzawa, IPDPS’06]
Super-Stabilization
[Dolev & Herman, CJTCS’97]
Retraite REGAL 2017, Cap Hornu

36. Retraite REGAL 2017, Cap Hornu
Related Work
Self-Stab + Safe Convergence
[Kakugawa & Masuzawa, IPDPS’06]
Super-Stabilization
[Dolev & Herman, CJTCS’97]
Retraite REGAL 2017, Cap Hornu

37. Back to Self-Stabilization
Retraite REGAL 2017, Cap Hornu

38. Back to Self-Stabilization
No safety guarantee
Retraite REGAL 2017, Cap Hornu

39. Back to Self-Stabilization
Ω(D)
Retraite REGAL 2017, Cap Hornu

40. Back to Self-Stabilization
Are all illegitimate configurations identically bad ?
Retraite REGAL 2017, Cap Hornu

41. Back to Self-Stabilization
Are all illegitimate configurations identically bad ?
Of course, NO !
Retraite REGAL 2017, Cap Hornu

42. Self-Stabilization + Safe Convergence
Really bad
No so bad
good
Retraite REGAL 2017, Cap Hornu

43. Self-Stabilization + Safe Convergence
Quick convergence time
Retraite REGAL 2017, Cap Hornu

44. Self-Stabilization + Safe Convergence
Optimal LC ⊆ feasable LC
Set of feasable LC: CLOSED
Set of optimal LC: CLOSED
Quick convergence to a feasable LC
(O(1) expected)
Convergence to an optimal LC
Retraite REGAL 2017, Cap Hornu

45. Retraite REGAL 2017, Cap Hornu
Related Work
Self-Stab + Safe Convergence
[Kakugawa & Masuzawa, IPDPS’06]
Super-Stabilization
[Dolev & Herman, CJTCS’97]
Retraite REGAL 2017, Cap Hornu

46. Retraite REGAL 2017, Cap Hornu
Super-Stabilization
Transient faults
Single topological change
Configurations of the system
time
Stabilization time
Superstabilization time
Retraite REGAL 2017, Cap Hornu

47. Super-Stabilization Quick convergence time Transient faults
Single topological change
Quick convergence time
Configurations of the system
time
Stabilization time
Superstabilization time
Retraite REGAL 2017, Cap Hornu

48. Retraite REGAL 2017, Cap Hornu
Super-Stabilization
Transient faults
Single topological change
Passage predicate
Configurations of the system
time
Stabilization time
Superstabilization time
Retraite REGAL 2017, Cap Hornu

49. Gradual Stabilization
Transient faults
𝜏 𝜚-dynamic steps
Quality of the specification
Configurations of the system T1
time
Stabilization time T T2 T3
Retraite REGAL 2017, Cap Hornu
T4 ≤ T

50. Case Study: Unison in Anonymous Networks
3 variants of the problem
Strong Unison
Weak Unison
Partial Unison
Each process has a local (logical) clock ∈ {0…𝛼-1}
Same liveness:
every process increments its clock infinitely often
But the safety is different
Retraite REGAL 2017, Cap Hornu

51. Retraite REGAL 2017, Cap Hornu
Safety 1 2
Strong
Weak
Partial 1 2 3 1 2 3 ?
Retraite REGAL 2017, Cap Hornu

52. Retraite REGAL 2017, Cap Hornu
Contribution
Transient faults
1 𝜚-dynamic step
Immediate
(Passage predicate)
Partial
Quality of the specification
Configurations of the system
Weak
Strong
1 round
time
Stabilization time Dn rounds
Retraite REGAL 2017, Cap Hornu
Dn rounds

53. Retraite REGAL 2017, Cap Hornu
𝜚-dynamic step?
The network remains Connected (necessary)
𝛼 > 3 ⇒ the network is Under Local Control (ULC)
ULC
ULC 1 2 3 1 2 3 ? ? ? ?
Retraite REGAL 2017, Cap Hornu

54. Retraite REGAL 2017, Cap Hornu
ULC necessary ?
𝛼 < 4 : NO
𝛼 = 4: Maybe (we have pathological cases)
𝛼 = 5: Maybe (we have pathological cases)
𝛼 > 5: YES
Retraite REGAL 2017, Cap Hornu

55. Gradual Stabilization: Perspectives
First attempt
𝜚 > 1 ?
Other (dynamic) problem
A good alternative to super-stabilization
Retraite REGAL 2017, Cap Hornu

56. Retraite REGAL 2017, Cap Hornu
References
[AD2017j] Karine Altisen, Stéphane Devismes, On probabilistic snap-stabilization, Theoretical Computer Science, Volume 688, 2017, Pages 49-76, ISSN
[ADDP16c] Karine Altisen, Stéphane Devismes, Anaïs Durand, and Franck Petit. Gradual Stabilization under T-Dynamics. Euro-Par 2016, 22nd International European Conference on Parallel and Distributed Computing. Pages , Grenoble (France), August 2016.
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57. Retraite REGAL 2017, Cap Hornu
Thank you!
Retraite REGAL 2017, Cap Hornu