1. Self-stabilizing (f,g)-Alliances with Safe Convergence Fabienne Carrier Ajoy K. Datta Stéphane Devismes Lawrence L. Larmore Yvan Rivierre

2. Co-Autors Ajoy K. Datta & Lawrence L. Larmore Fabienne Carrier & Yvan Rivierre 14/11/13SSS 2013, Osaka

3. Roadmap 1.Safe convergence 2.The (f,g)-alliance problem 3.Contribution 4.Algorithm 5.Perspectives 14/11/13SSS 2013, Osaka

4. Safe convergence 14/11/13SSS 2013, Osaka

5. Pros and Cons of Self-Stabilization Tolerate any finite number of transient faults No initialization – Large-scale network – Self-organization in sensor network Dynamicity – Topological change ≈ Transient fault Tolerate only transient faults Eventual safety No stabilization detection 14/11/13SSS 2013, Osaka

6. Pros and Cons of Self-Stabilization Tolerate any finite number of transient faults No initialization – Large-scale network – Self-organization in sensor network Dynamicity – Topological change ≈ Transient fault Tolerate only transient faults Eventual safety No stabilization detection 14/11/13SSS 2013, Osaka

7. Related Work Enhancing safety: – Fault-containment [Ghosh et 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. 14/11/13SSS 2013, Osaka

8. Back to self-stabilization 14/11/13SSS 2013, Osaka

9. Back to self-stabilization No safety guarantee 14/11/13SSS 2013, Osaka

10. Back to self-stabilization Ω(D) 14/11/13SSS 2013, Osaka

11. Back to self-stabilization Are all illegitimate configurations identically bad ? 14/11/13SSS 2013, Osaka

12. Back to self-stabilization Are all illegitimate configurations identically bad ? Of course, NO ! 14/11/13SSS 2013, Osaka

13. Self-stabilization + Safe Convergence Not so bad Really bad good 14/11/13SSS 2013, Osaka

14. Self-stabilization + Safe Convergence Quick convergence time 14/11/13SSS 2013, Osaka

15. 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 14/11/13SSS 2013, Osaka

16. Self-stabilization + Safe Convergence: example [Kakugawa & Masuzawa, IPDPS’06] Construction of a minimal dominating set – 1-round convergence to a dominating set (not necessarily a minimal one) – Then, O(D)-rounds convergence to a MINIMAL dominating set During this phase, all configurations contain a dominating set 14/11/13SSS 2013, Osaka

17. The problem: (f,g)-Alliance [Dourado et al, SSS’11] Alliance: subset of nodes f, g: 2 functions mapping nodes to natural integers For every process p: – p ∉ Alliance ⇒ at least f(p) neighbors ∈ Alliance – p ∈ Alliance ⇒ at least g(p) neighbors ∈ Alliance 14/11/13SSS 2013, Osaka

18. Example: (f,g)-Alliance Red nodes form a (1,0)-Alliance 14/11/13SSS 2013, Osaka

19. Example: (f,g)-Alliance Red nodes DO NOT form a (1,0)-Alliance 14/11/13SSS 2013, Osaka

20. (f,g)-Alliance: generalization of several problems Dominating sets K-domination sets K-tuple domination sets Global defensive alliance Global offensive alliance E.g., Dominating set = (1,0)-alliance 14/11/13SSS 2013, Osaka

21. Minimality & 1-Minimality Let A be a set of nodes A is a minimal (f,g)-Alliance iff every 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 14/11/13SSS 2013, Osaka

22. Example: (0,1)-Alliance Red nodes form a (0,1)-Alliance, but NEITHER a minimal NOR a 1-minimal (0,1)-Alliance 14/11/13SSS 2013, Osaka

23. Example: (0,1)-Alliance Red nodes form a 1-minimal (0,1)-Alliance but not a minimal one 14/11/13SSS 2013, Osaka

24. Example: (0,1)-Alliance Red nodes (empty set) both form a minimal AND a 1-minimal (0,1)-Alliance 14/11/13SSS 2013, Osaka

25. Property [Dourado et 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 14/11/13SSS 2013, Osaka

26. Contribution Self-Stabilizing Safe Converging Algorithm for computing: a 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(log n) bits per process Step complexity: O(Δ 3 n) 14/11/13SSS 2013, Osaka

27. Algorithm’s main ideas 14/11/13SSS 2013, Osaka

28. ``Naïve Idea” One Boolean Red: ∈ A Green: ∉ A Two actions: Join Leave 14/11/13SSS 2013, Osaka

29. ``Naïve Idea” One boolean Red: ∈ A Green: ∉ A Two actions: Join Leave To obtain safe convergence, it should be harder to leave than to join 14/11/13SSS 2013, Osaka

30. Leave the alliance p can leave if : 1.At least f(p) neighbors ∈ A after p leaves AND 2.Each neighbor q still have enough neighbors ∈ A after p leaves i.e., g(q) or f(q) depending whether q belongs or not to A 14/11/13SSS 2013, Osaka

31. At least f(p) neighbors ∈ A after p leaves Leaving should be locally sequential Example: (2,1)-Alliance p 14/11/13SSS 2013, Osaka

32. At least f(p) neighbors ∈ A after p leaves Leaving should be locally sequential Example: (2,1)-Alliance p 14/11/13SSS 2013, Osaka p

33. At least f(p) neighbors ∈ A after p leaves Leaving should be locally sequential Example: (2,1)-Alliance p 14/11/13SSS 2013, Osaka

34. At least f(p) neighbors ∈ A after p leaves Leaving should be locally sequential Example: (2,1)-Alliance p 14/11/13SSS 2013, Osaka p

35. Pointer: authorization to leave Nil 14/11/13SSS 2013, Osaka p Nil

36. Pointer: authorization to leave Nil 14/11/13SSS 2013, Osaka p Nil

37. Each neighbor still have enough neighbor ∈ A after p leaves A neighbor q gives an authorization only if q still have enough neighbors ∈ A without p p q (1,0)-Alliance 14/11/13SSS 2013, Osaka

38. Each neighbor still have enough neighbor ∈ A after p leaves A neighbor q gives an authorization only if q still have enough neighbors ∈ A without p p q (1,0)-Alliance If q has several choices ID breaks ties 14/11/13SSS 2013, Osaka

39. Nil Deadlock problems Nil (1,0)-Alliance 14/11/13SSS 2013, Osaka

40. Nil Deadlock problems Nil (1,0)-Alliance Busy! 14/11/13SSS 2013, Osaka

41. Deadlock problems Nil (1,0)-Alliance Busy! Nil 14/11/13SSS 2013, Osaka

42. Deadlock problems Nil (1,0)-Alliance Busy! Tie break! Nil 14/11/13SSS 2013, Osaka

43. Deadlock problems Nil (1,0)-Alliance Busy! Tie break! Nil 14/11/13SSS 2013, Osaka

44. Deadlock problems Nil (1,0)-Alliance Busy! Tie break! Nil 14/11/13SSS 2013, Osaka

45. How to evaluate Busy? N P ∩ A < f(p) (2,0)-Alliance p Busy! 14/11/13SSS 2013, Osaka

46. How to evaluate Busy? N P ∩ A < f(p) A neighbor q of p needs that p stays in the alliance (2,0)-Alliance p q Busy! 14/11/13SSS 2013, Osaka

47. 0 How to evaluate Busy? N P ∩ A < f(p) A neighbor q of p needs that p stays in the alliance 23 (2,0)-Alliance 2 p q Busy! 1 1 Nb 14/11/13SSS 2013, Osaka 22

48. Last problem … (1,0)-Alliance Nil 14/11/13SSS 2013, Osaka

49. Last problem … (1,0)-Alliance Nil 14/11/13SSS 2013, Osaka

50. Last problem … (1,0)-Alliance Nil 14/11/13SSS 2013, Osaka

51. Last problem … Solution: strict alternation Nil, Nil 14/11/13SSS 2013, Osaka

52. Last problem … Solution: strict alternation Nil, Nil 14/11/13SSS 2013, Osaka

53. Join the Alliance p ∉ A and N P ∩ A < f(p) (EASY) A neighbor q needs that p joins the alliance: – Evaluated by reading the status of q and q.Nb No neighbor points to p. 14/11/13SSS 2013, Osaka

54. Perspectives (Total) Stabilization in O(D) using O(log n) bits? Self-stabilization with safe convergence without any assumption on f and g? 14/11/13SSS 2013, Osaka

55. Thank you! 14/11/13SSS 2013, Osaka

56. At least f(p) neighbors ∈ A after p leaves Leaving should be locally sequential Example: (2,2)-Alliance p Neighbors 14/11/13SSS 2013, Osaka

57. At least f(p) neighbors ∈ A after p leaves Leaving should be locally sequential Example: (2,2)-Alliance p Neighbors p 14/11/13SSS 2013, Osaka

58. At least f(p) neighbors ∈ A after p leaves Leaving should be locally sequential Example: (2,2)-Alliance p Neighbors p 14/11/13SSS 2013, Osaka

59. Pointer: authorization to leave Nil p Neighbors 14/11/13SSS 2013, Osaka