1. A Snap-Stabilizing DFS with a Lower Space Requirement
SSS’2005, October , Barcelona (Spain)
A Snap-Stabilizing DFS with a Lower Space Requirement
Alain Cournier, Stéphane Devismes, and Vincent Villain

2. Snap-Stabilizing System?
A particular class of self-stabilizing systems.
(A self-stabilizing system, regardless of the initial state of the processors, is guaranteed to converge to the intended behavior in finite time. (Dijkstra 1974))
A snap-stabilizing system, regardless of the initial state of the processors, always behaves according to its specifications. (Bui et al, 1999)
A Snap-Stabilizing DFS with a Lower Space Requirement

3. A Snap-Stabilizing DFS with a Lower Space Requirement
Model R
Arbitrary rooted network
State model
Local shared memory
Daemon: weakly fair/unfair
A Snap-Stabilizing DFS with a Lower Space Requirement

4. O(NN) states (Ids List)
Related Works
Self-stabilizing area:
Huang and Chen (Distributed Computing, 1993)
Johnen and Beauquier (WSS, 1995)
Johnen et al (WDAG, 1997)
Datta et al (Distributed Computing, 2000)
Snap-stabilizing area:
Cournier et al, Transformer (ICDCS, 2003)
Cournier et al (OPODIS, 2004)
Weakly fair Daemon
Unfair Daemon
O(NN) states (Ids List)
A Snap-Stabilizing DFS with a Lower Space Requirement

5. A Snap-Stabilizing DFS with a Lower Space Requirement
Write a snap-stabilizing unfair DFS protocol with a lower memory requirement (no Ids)
A Snap-Stabilizing DFS with a Lower Space Requirement

6. DFS in a non-faulty configuration
A Snap-Stabilizing DFS with a Lower Space Requirement

7. Problem of the cleaning by the leaves
Visited for the 2nd time
A Snap-Stabilizing DFS with a Lower Space Requirement

8. Problem of the cleaning by the leaves
...
...
A Snap-Stabilizing DFS with a Lower Space Requirement

9. Solution: cleaning at the end of the traversal only
A Snap-Stabilizing DFS with a Lower Space Requirement

10. Insuring the visit of all the processors
A Snap-Stabilizing DFS with a Lower Space Requirement

11. Insuring the visit of all the processors
A Snap-Stabilizing DFS with a Lower Space Requirement

12. Insuring the visit of all the processors
Processors never visited r
Not Snap-stabilizing!
A Snap-Stabilizing DFS with a Lower Space Requirement

13. Solution: a Question Mecanism
[Blin et al, An Improved Snap-Stabilizing PIF Algorithm, 2003]
Can I switch to the red color?
Only the root can deliver an answer and this answer is propagated in the spanning tree of the root only
A Snap-Stabilizing DFS with a Lower Space Requirement

14. Insuring the visit of all the processorsOk Ok Ok Ok r Ok Ok Ok
A Snap-Stabilizing DFS with a Lower Space Requirement

15. Insuring the visit of all the processorsOk Ok Ok
The yellow processor waits
Until the abnormal traversal disappears by Error Correction
A Snap-Stabilizing DFS with a Lower Space Requirement

16. A Snap-Stabilizing DFS with a Lower Space Requirement
Error Correction 1 7 2 8 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

17. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

18. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

19. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

20. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

21. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

22. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

23. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

24. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

25. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

26. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

27. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

28. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

29. Solution: Paralyzing Mechanism1 2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

30. Solution: Paralyzing Mechanism2 3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

31. Solution: Paralyzing Mechanism3 6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

32. Solution: Paralyzing Mechanism6 R 5 4
A Snap-Stabilizing DFS with a Lower Space Requirement

33. Solution: Paralyzing Mechanism6 R 5
A Snap-Stabilizing DFS with a Lower Space Requirement

34. Solution: Paralyzing Mechanism6 R
A Snap-Stabilizing DFS with a Lower Space Requirement

35. Solution: Paralyzing Mechanism
A Snap-Stabilizing DFS with a Lower Space Requirement

36. A Snap-Stabilizing DFS with a Lower Space Requirement
Conclusion
Snap-Stabilizing
Arbitrary Rooted Networks
Unfair Daemon
A Snap-Stabilizing DFS with a Lower Space Requirement

37. Conclusion: SSS’05 vs OPODIS’04
Memory Requirement
O(NN)
O(∆²× N)
Ids
Yes No
Stabilization time
Delay (rounds)
O(N)
O(N²)
Delay (steps)
O(N3)
Execution (rounds)
Execution (steps)
In fact, our protocol stabilizes in 0 times but the complexity of the delay to start a token circulation as well as the complexity for execute a complete token circulation are worst than the result obtained with the protocol of Cournier and al.
A Snap-Stabilizing DFS with a Lower Space Requirement

38. Perspective
Finding an efficient solution in both time and space complexities
A Snap-Stabilizing DFS with a Lower Space Requirement

39. A Snap-Stabilizing DFS with a Lower Space Requirement
Thank You!
A Snap-Stabilizing DFS with a Lower Space Requirement