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

2. Skip Chain October, 11, 2011SSS'2011, Grenoble2 Left Right Major nodes Relay nodes

3. Skip Chain Sorting October, 11, 2011SSS'2011, Grenoble3 3 3 6 6 2 2 2 2 3 3 6 6

4. 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 October, 11, 2011SSS'2011, Grenoble4

5. Overview Idea : distributed bubble sort October, 11, 2011SSS'2011, Grenoble5 Arbitrary Configurations Normal Configurations Legitimate Configurations Error Correction Sorting

6. Data Structure October, 11, 2011SSS'2011, Grenoble6 3 3 2 2 6 6

7. Swap October, 11, 2011SSS'2011, Grenoble7 2 2 6 6 6 6 6 6 6 6 2 2 6 6 6 6 2 2 2 2 2 2 6 6 2 2 2 2 Synchronization between swaps : 4 colors

8. 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 October, 11, 2011SSS'2011, Grenoble8

9. Example October, 11, 2011SSS'2011, Grenoble9 X Yz 3 3 3 3 0 0 3 3 2 2 0 0 Compare and swap V(y)V(x)

10. Example October, 11, 2011SSS'2011, Grenoble10 X Yz 3 3 3 3 1 1 0 0 2 2 0 0 V(y)’V(x)’ 0, V(x)’1, V(y)’

11. Example October, 11, 2011SSS'2011, Grenoble11 X Yz 0 0 3 3 1 1 0 0 2 2 1 1 Compare and swap V(y)’V(x)’ V(y)’ V(u) V(x)’

12. Example October, 11, 2011SSS'2011, Grenoble12 X Yz 1 1 0 0 1 1 0 0 3 3 2 2 Compare and swap V(y)’V(x)’ V(x)’’ V(y)’ V(u)’ 2 1, V(x)’’

13. Example October, 11, 2011SSS'2011, Grenoble13 X Yz 1 1 0 0 2 2 1 1 3 3 2 2 V(y)’V(x)’’ V(y)’ V(u)’

14. Example October, 11, 2011SSS'2011, Grenoble14 X Yz 1 1 0 0 3 3 2 2 3 3 2 2 V(y)’V(x)’’ V(y)’ V(u)’ 2 3

15. Example October, 11, 2011SSS'2011, Grenoble15 X Yz 2 2 0 0 3 3 2 2 3 3 3 3 V(y)’V(x)’’ V(y)’ V(u)’

16. Example October, 11, 2011SSS'2011, Grenoble16 X Yz 2 2 1 1 3 3 2 2 0 0 3 3 V(y)’V(x)’’ V(y)’ V(z) V(u)’ Compare and swap

17. Example October, 11, 2011SSS'2011, Grenoble17 X Yz 3 3 2 2 3 3 2 2 1 1 0 0 V(y)’V(x)’’ V(y)’’ V(z)’ V(u)’ 0, V(y)’’ 3

18. Example October, 11, 2011SSS'2011, Grenoble18 X Yz 3 3 2 2 0 0 3 3 1 1 0 0 V(y)’’V(x)’’ V(y)’’ V(z)’ V(u)’

19. Example October, 11, 2011SSS'2011, Grenoble19 X Yz 3 3 3 3 0 0 3 3 2 2 0 0 V(y)’’V(x)’’ V(y)’’ V(z)’ V(u)’

20. Error correction October, 11, 2011SSS'2011, Grenoble20

21. Silence October, 11, 2011SSS'2011, Grenoble21 2 2 2 2 3 3 3 3 3 3 3 3 6 6 2 2 3 3 2 2 Done

22. Perspective Can we enhance the round complexity to O(n) rounds ? Step complexity ? October, 11, 2011SSS'2011, Grenoble22

23. Thank you October, 11, 2011SSS'2011, Grenoble23

24. Min-Max Search Tree October, 11, 2011SSS'2011, Grenoble24 10 1 1 9 9 5 5 8 8 4 4 7 7 6 6 2 2 <= 3 3 min max