1. Snap-Stabilizing Depth-First Search on Arbitrary Networks Alain Cournier, Stéphane Devismes, Franck Petit, and Vincent Villain OPODIS 2004, December 15-17 2004, Grenoble (France)

2. What is Depth-First Search? Root R

3. Applications of Depth-First Search? Mutual Exclusion Spanning Tree Computation Constraint Programming Routing …

4. Self- and Snap- Stabilisation A self-stabilizing system, regardless of the initial state of the processors, is guaranteed to converge to the intended behavior in finite time. Self-Stabilisation, Dijkstra (1974) A snap-stabilizing system, regardless of the initial state of the processors, always behaves according to its specifications. Snap-Stabilisation Bui, Datta, Petit, and Villain (1999) Remark: Self-Stabilisation is a general technique allowing to design systems tolerating transient failures.

5. More Precisely… Let T be a task and SPt a specification of T. Let P be a protocol. P is snap-stabilizing for SPt iff: 1.A particular action of P will be executed, 2.The result obtained from this particular action satisfies SPt.

6. State Model Communication: Local Shared Memory Computation: Distributed Fairness: Weakly fair, Unfair Time Complexity: Step and Round

7. Related Works Self-Stabilisation Area: Huang and Chen (1993) Johnen and Beauquier (1995) Petit and Villain (1997) Datta, Johnen, Petit, and Villain (2000) Petit (2001)

8. Related Works Snap-Stabilisation Area: Petit and Villain (1999) for tree networks Cournier, Datta, Petit, and Villain (2003), Transformer

9. There does not exist any solution using an unfair daemon

10. Our Solution

11. Identified Rooted Networks 8 123 4 5 6 9 11 10 12 13 14 15 16 17 18 7 R

12. Data Structures Successor pointor: Parent pointor: Visited Set: {1,2,3} IdlePointingDone

13. 12 3 8 9 5 From a « good » initial configuration {5} {5,6} {5,6,7} {5,6,7,3,2,4,8,9,1} {5,6,7,3,2} 6 7 {5,6,7,3,2,4} 4 {5,6,7,3,2,4,8} {5,6,7,3} {5,6,7,3,2,4,8} {5,6,7,3,2} {5,6,7,3,2,4,8} {5,6,7,3,2,4,8,9} {5,6,7,3,2,4,8,9,1} R

14. 123 4 6 9 10 11 12 13 14 15 17 18 {7}{7,8} {5,7} {5,7,10} {5,7,10,15} Arbitrary Configuration 7 16 5 8 {3,4} {2,3} {6,8} {6,8,11} Done Processor Abnormal Traversals R

15. How to correct Abnormal Traversals?

16. Error Detection 7 5 8 9 {6}{6,7} 6 {6,7,8} Abnormal Root

17. Error Correction Cleaning the successive Abnormal Root of each Abnormal Traversal

18. 123 4 6 11 12 13 14 15 17 {5,7} {5,7,10} {5,7,10,15} 7 16 5 8 {2,3} {6,8} {6,8,11} {3,4} {5,7,10,15,17} 18 {5,7,10,15,17,18} {5,7,10,15,17,18,16} {7} {5,7,10,15,17,18,16} Starting Action {7,8} {5,7,10,15,17,18,16} {7,8,9} 9 {7,8,9,16} 10 The Root is now enabled to initiates the protocol Snap? R

19. 123 4 6 11 12 13 14 15 17 {5,7} {5,7,10} {5,7,10,15} 7 16 5 8 {3,4} {5,7,10,15,17} 18 {5,7,10,15,17,18}{5,7,10,15,17,18,16} {7} {7,8} {5,7,10,15,17,18,16} 9 10 Unfairness? R

20. Conclusion Snap-Stabilizing Protocol for Arbitrary Networks Works under an Unfair Daemon Space Requirement: O(N ×log(N)) per processors Time Complexity: Delay: O(N²) steps, O(N) rounds DFS-Wave: O(N²) steps, O(N) rounds

21. Perspectives Finding a solution with a lower memory requirement Finding a Transformer working with an unfair daemon Finding a solution without using ids