1. Sorting on Skip Chains Ajoy K. Datta, Lawrence L. Larmore, and Stéphane Devismes

2. Skip Chain December 2, 2011PDAA'2011, Osaka Left Right Major nodes Relay nodes

3. Skip Chain Sorting December 2, 2011PDAA'2011, Osaka 3 6 2

4. Skip Chain Sorting December 2, 2011PDAA'2011, Osaka 3 6 2 2 3 6

5. Contribution Skip Chain Sorting Algorithm – Self-stabilizing – Silent – Locally shared memory model Unfair demon O(b) space, b = number of bits to encode a value O(md) rounds – m : number of major nodes – d : maximum number of relay between two major nodes – md = O(n) if the spacing between major processes is roughtly equal December 2, 2011PDAA'2011, Osaka

6. December 2, 2011PDAA'2011, Osaka Self-Stabilization: Closure + Convergence

7. December 2, 2011PDAA'2011, Osaka Self-Stabilization: Closure + Convergence States of the System

8. December 2, 2011PDAA'2011, Osaka Self-Stabilization: Closure + Convergence States of the System Illegitimate States Legitimate States

9. December 2, 2011PDAA'2011, Osaka Self-Stabilization: Closure + Convergence States of the System Closure

10. December 2, 2011PDAA'2011, Osaka Self-Stabilization: Closure + Convergence States of the System Closure

11. December 2, 2011PDAA'2011, Osaka Self-Stabilization: Closure + Convergence States of the System Closure

12. December 2, 2011PDAA'2011, Osaka Self-Stabilization: Closure + Convergence States of the System Convergence

13. December 2, 2011PDAA'2011, Osaka Self-Stabilization: Closure + Convergence States of the System Convergence

14. December 2, 2011PDAA'2011, Osaka Self-Stabilization: Closure + Convergence States of the System Convergence

15. December 2, 2011PDAA'2011, Osaka Tolerate Transient Faults E.g., any finite number of memory or message corruptions, or topological changes

16. December 2, 2011PDAA'2011, Osaka Tolerate Transient Faults E.g., any finite number of memory or message corruptions, or topological changes

17. Silence “The system eventually reaches a terminal configuration where the values of the variables are fixed” [Dolev et al, 1996] December 2, 2011PDAA'2011, Osaka

18. Related Work New problem A generalization of the sorting problem – Oriented chain [Bein et al, 2008] Self-stabilizing O(n) round, O(b) space December 2, 2011PDAA'2011, Osaka

19. Overview December 2, 2011PDAA'2011, Osaka

20. Overview Idea : distributed bubble sort December 2, 2011PDAA'2011, Osaka

21. Overview Idea : distributed bubble sort December 2, 2011PDAA'2011, Osaka Arbitrary Configurations

22. Overview Idea : distributed bubble sort December 2, 2011PDAA'2011, Osaka Arbitrary Configurations Normal Configurations Error Correction

23. Overview Idea : distributed bubble sort December 2, 2011PDAA'2011, Osaka Arbitrary Configurations Normal Configurations Legitimate Configurations Error Correction Sorting

24. Data Structure December 2, 2011PDAA'2011, Osaka 3 6 2

25. Swap December 2, 2011PDAA'2011, Osaka 6 2

26. Swap December 2, 2011PDAA'2011, Osaka 6 2 6 266 6 6

27. Swap December 2, 2011PDAA'2011, Osaka 6 2 6 266 6 6 2 622 2 2

28. Swap December 2, 2011PDAA'2011, Osaka 6 2 6 266 6 6 2 622 2 2 Synchronization between swaps : 4 colors

29. Colors A value moves to the left at the crest of wave 0 A value moves to the right at the crest of wave 1 Colors 2 and 3 to avoid ambiguïty and to synchronize Color E: error color December 2, 2011PDAA'2011, Osaka

30. Example December 2, 2011PDAA'2011, Osaka X YZ 3 3 3 3 0 0 3 3 2 2 0 0 Compare and swap V(y)V(x) 3 3 3 3 3 3 0 0 0 0 0 0

31. Example December 2, 2011PDAA'2011, Osaka X YZ 3 3 3 3 1 1 0 0 2 2 0 0 V(y)’V(x)’ 3 3 3 3 3 3 0 0 0 0 0 0 0, V(x)’1, V(y)’

32. Example December 2, 2011PDAA'2011, Osaka X YZ 0 0 3 3 1 1 0 0 2 2 1 1 V(y)’V(x)’ 0 0 0 0 0 0 1 1 1 1 1 1 Compare and swap V(y)’V(u) V(x)’

33. Example December 2, 2011PDAA'2011, Osaka X YZ 1 1 0 0 1 1 0 0 3 3 2 2 V(y)’V(x)’ 0 0 0 0 0 0 1 1 1 1 1 1 V(y)’V(u)’ V(x)’’ 2 1, V(x)’’

34. Example December 2, 2011PDAA'2011, Osaka X YZ 1 1 0 0 2 2 1 1 3 3 2 2 V(y)’V(x)’’ 1 1 1 1 1 1 2 2 2 2 2 2 V(y)’V(u)’ V(x)’’

35. Example December 2, 2011PDAA'2011, Osaka X YZ 1 1 0 0 3 3 2 2 3 3 2 2 V(y)’V(x)’’ 1 1 1 1 1 1 2 2 2 2 2 2 V(y)’V(u)’ V(x)’’ 2 3

36. Example December 2, 2011PDAA'2011, Osaka X YZ 2 2 0 0 3 3 2 2 3 3 3 3 V(y)’V(x)’’ 2 2 2 2 2 2 3 3 3 3 3 3 V(y)’V(u)’ V(x)’’ 0, V(z) 1, V(u)’’

37. Example December 2, 2011PDAA'2011, Osaka X YZ 2 2 1 1 3 3 2 2 0 0 3 3 V(y)’V(x)’’ 2 2 2 2 2 2 3 3 3 3 3 3 V(y)’ V(z)V(u)’’ V(x)’’ Compare and swap

38. Example December 2, 2011PDAA'2011, Osaka X YZ 3 3 2 2 3 3 2 2 1 1 0 0 V(y)’V(x)’’ 2 2 2 2 2 2 3 3 3 3 3 3 V(y)’’ V(z)’V(u)’’ V(x)’’ 0, V(y)’’ 3

39. Example December 2, 2011PDAA'2011, Osaka X YZ 3 3 2 2 0 0 3 3 1 1 0 0 V(y)’’V(x)’’ 3 3 3 3 3 3 0 0 0 0 0 0 V(y)’’ V(z)’V(u)’’ V(x)’’ 3 2

40. Example December 2, 2011PDAA'2011, Osaka X YZ 3 3 3 3 0 0 3 3 2 2 0 0 V(y)’’V(x)’’ 3 3 3 3 3 3 0 0 0 0 0 0 V(y)’’ V(z)’V(u)’’ V(x)’’

41. Error correction December 2, 2011PDAA'2011, Osaka

42. Error correction December 2, 2011PDAA'2011, Osaka

43. Error correction December 2, 2011PDAA'2011, Osaka

44. Error correction December 2, 2011PDAA'2011, Osaka

45. Error correction December 2, 2011PDAA'2011, Osaka

46. Uncorrect swap December 2, 2011PDAA'2011, Osaka X YZ 3 3 3 3 0 0 3 3 2 2 0 0 Compare and swap bc 3 3 3 3 3 3 0 0 0 0 0 0 a c c abbbb

47. Uncorrect swap December 2, 2011PDAA'2011, Osaka X YZ 3 3 3 3 1 1 0 0 2 2 0 0 3 3 3 3 3 3 0 0 0 0 0 0 0, b1, c Bad swap cb a c c abbbb

48. Uncorrect swap December 2, 2011PDAA'2011, Osaka X YZ 3 3 3 3 1 1 0 0 2 2 0 0 3 3 3 3 3 3 0 0 0 0 0 0 0, b1, c cb a c c abbbb a Error correction

49. Uncorrect swap December 2, 2011PDAA'2011, Osaka X YZ 3 3 3 3 1 1 0 0 2 2 0 0 3 3 3 3 3 3 1 1 1 1 0 0 1, c cb a a a accbb a Error correction

50. Uncorrect swap December 2, 2011PDAA'2011, Osaka X YZ 3 3 3 3 1 1 0 0 2 2 0 0 3 3 3 3 3 3 1 1 1 1 0 0 1, c cb a a a accbb a Error correction To undo the swap, we should remember that we do a swap! Status : S or U S

51. Uncorrect swap December 2, 2011PDAA'2011, Osaka X YZ 3 3 3 3 1 1 0 0 2 2 0 0 3 3 3 3 3 3 1 1 1 1 0 0 1, c ba a a a accbb b Error correction To undo the swap, we should remember that we do a swap! Status : S or U S

52. Silence December 2, 2011PDAA'2011, Osaka An additional Boolean variable : Done Done => Stop initiating wave color 2222333336

53. Silence December 2, 2011PDAA'2011, Osaka An additional Boolean variable : Done Done => Stop initiating wave color 2222333336

54. Silence December 2, 2011PDAA'2011, Osaka An additional Boolean variable : Done Done => Stop initiating wave color 2222333336 3≤6 & Color=3 → Done<-True; Color<-0

55. Silence December 2, 2011PDAA'2011, Osaka 2222333336 Color=3 & R.color=0 & R.done → Done<-True; Color<-0 Value<-R.Value

56. Silence December 2, 2011PDAA'2011, Osaka 2222333336 Color=3 & R.color=0 & R.done & Value≤R.Value → Done<-True; Color<-0

57. Silence December 2, 2011PDAA'2011, Osaka 2222333336 At the end, for all node: - Color = 0 - Done = true No enabled action

58. Perspective Can we enhance the round complexity to O(n) rounds ? Step complexity ? December 2, 2011PDAA'2011, Osaka

59. Thank you December 2, 2011PDAA'2011, Osaka

60. Min-Max Search Tree December 2, 2011PDAA'2011, Osaka 10 1 1 9 9 5 5 8 8 3 3 7 7 6 6 2 2 <= 4 4 min max