Self-Stabilizing Algorithm with S afe C onvergence building an ( f , g )-Alliance

1. Self-Stabilizing Algorithm with Safe Convergence building an (f,g)-Alliance Fabienne Carrier Ajoy K. Datta Stéphane Devismes Lawrence L. Larmore YvanRivierre

2. Co-Autors Ajoy K. Datta & Lawrence L. Larmore Fabienne Carrier & YvanRivierre

3. Roadmap • Self-stabilization • Safe convergence • The problem • Contribution • Algorithm • Perspectives

4. Self-Stabilization [Dijkstra,74]

5. Self-Stabilization [Dijkstra,74] Transient Faults, e.g., memory corruption, message lost …

6. Self-Stabilization [Dijkstra,74]

7. Self-Stabilization [Dijkstra,74]

8. Self-Stabilization [Dijkstra,74]

9. Self-Stabilization [Dijkstra,74]

10. Self-Stabilization [Dijkstra,74]

11. Self-Stabilization [Dijkstra,74] Recover after any number of transient faults

12. Pros and Cons • Tolerate any finite number of transient faults • No initialization • Large-scale network • Self-organization in sensor network • Dynamicity • Topological change ≈ Transient fault • Tolerate onlytransient faults • Eventual safety • No stabilization detection

13. Pros and Cons • Tolerate any finite number of transient faults • No initialization • Large-scale network • Self-organization in sensor network • Dynamicity • Topological change ≈ Transient fault • Tolerate onlytransient faults • Eventual safety • No stabilization detection

14. Related Work • Enhancing safety: • Fault-containment [Ghoshet al, PODC’96] • Superstabilization[Dolev & Herman, CJTCS’97] • Time-adaptive Self-stabilization [Kutten & Patt-Shamir, PODC’97] • Self-Stab + safe convergence [Kakugawa & Masuzawa, IPDPS’06] • Etc.

15. Back to self-stabilization

16. Back to self-stabilization No safety guarantee

17. Back to self-stabilization Ω(D)

18. Back to self-stabilization Are all illegitimate configurations identically bad ?

19. Back to self-stabilization Are all illegitimate configurations identically bad ? Of course, NO !

20. Self-stabilization + Safe Convergence Really bad No so bad good

21. Self-stabilization + Safe Convergence Quick convergence time

22. 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

23. The problem: (f,g)-Alliance[Douradoet al, SSS’11] • Alliance: subset of nodes • f, g: 2 functions mapping nodes to natural integers • For every process p: • p ∉ Alliance ⇒ at leastf(p) neighbors ∈ Alliance • p ∈ Alliance ⇒ at least g(p) neighbors ∈ Alliance

24. Example: (f,g)-Alliance Red nodes form a (1,0)-Alliance

25. Example: (f,g)-Alliance Red nodes DO NOT form a (1,0)-Alliance

26. (f,g)-Alliance: generalization of several problems • Dominating sets • K-dominating sets • K-tuple dominating sets • Global defensive alliance • Global offensive alliance

27. Minimality & 1-Minimality • Let A be a set of nodes • A is a minimal (f,g)-Alliance iffevery proper subset of A is not an (f,g)-Alliance • A is a 1-minimal (f,g)-Alliance iff ∀p ∈ A, A-{p} is not an (f,g)-Alliance

28. Example: (0,1)-Alliance Red nodes form NEITHER a minimal NOR a 1-minimal (0,1)-Alliance

29. Example: (0,1)-Alliance Red nodes form a 1-minimal (0,1)-Alliance but not a minimal one

30. Example: (0,1)-Alliance Red nodes (empty set) both form a minimal AND a 1-minimal (0,1)-Alliance

31. Property[Douradoet al, SSS’11] • Every minimal (f,g)-Alliance is a 1-minimal (f,g)-Alliance • If for every node p, f(p) ≥ g(p), then • A is a minimal (f,g)-Alliance iff A is a 1-minimal (f,g)-Alliance

32. Contribution • Self-Stabilizing Safe Converging Algorithm for computing the minimal (f,g)-Alliance in identified networks: • Safe Convergence • Stabilization in 4 rounds to a configuration, where an (f,g)-Alliance is defined • Stabilization in 4n+4 additional rounds to a configuration, where minimal (f,g)-Alliance is defined • Assumptions: • If for every node p, f(p) ≥ g(p) and δ(p) ≥ g(p) • Locally shared memory model, unfair daemon • Other complexities • Memory requirement: O(logn) bits per process • Step complexity: O(Δ3n) (number of state changes)

33. Locally Shared Memory Model

34. Locally Shared Memory Model

35. Locally Shared Memory Model

36. Locally Shared Memory Model Choices of the scheduler (daemon) Unfair daemon: no fairness assumption

37. Algorithm’s main ideas

38. ``Naïve Idea” • One boolean • Red: ∈ A • Green: ∉ A • Two actions: • Join • Leave

39. ``Naïve Idea” To obtain safe convergence, it should be harder to leave than to join • One boolean • Red: ∈ A • Green: ∉ A • Two actions: • Join • Leave

40. Leave the alliance • p can leaves if : • At least f(p) neighbors ∈ A after pleaves AND • Each neighbor still have enough neighbors ∈ A after pleaves

41. At least f(p) neighbors ∈ A after pleaves • Leaving should be locally sequential • Example: (2,2)-Alliance p Neighbors

42. At least f(p) neighbors ∈ A after pleaves • Leaving should be locally sequential • Example: (2,2)-Alliance p p Neighbors Neighbors

43. At least f(p) neighbors ∈ A after pleaves • Leaving should be locally sequential • Example: (2,2)-Alliance p p Neighbors Neighbors

44. Pointer: authorization to leave Nil p Neighbors

45. Each neighbor still have enough neighbor ∈ A after pleaves • A neighbor q gives an authorization only if q still have enough neighbors ∈ A without p p (1,0)-Alliance q

46. Each neighbor still have enough neighbor ∈ A after pleaves • A neighbor q gives an authorization only if q still have enough neighbors ∈ A without p p (1,0)-Alliance If q has several choices ID breaks ties q

47. Deadlock problems Nil Nil (1,0)-Alliance

48. Deadlock problems Busy ! Nil Nil (1,0)-Alliance

49. Deadlock problems Busy ! Nil Nil (1,0)-Alliance

50. Deadlock problems Busy ! Nil Nil Tie break ! (1,0)-Alliance