1. Snap-Stabilizing PIF and Useless Computations Alain Cournier, Stéphane Devismes, and Vincent Villain ICPADS’2006, July 12-15 2006, Minneapolis (USA)

2. Snap-stabilizing PIF and Useless Computations2 PIF scheme v PIF = Propagation of informations with feedback v Any processor can be the initiator of a PIF wave. v A PIF wave : –A processor r (root of this PIF wave) initiates the wave by broadcasting a message m. –The wave terminates at r and when that happens, all processors (  r) have acknowledged the receipt of m.

3. Snap-stabilizing PIF and Useless Computations3 Stabilizing systems v 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] v A snap-stabilizing system, regardless of the initial state of the processors, always behaves according to its specification. [Bui et al, 1999] v Application: these systems tolerates transient failure

4. Snap-stabilizing PIF and Useless Computations4 Self- vs Snap- stabilizing PIF v Self-stabilizing PIF: –Infinity of PIF waves. –After a finite number of PIF waves, the system recovers the intended behavior. v Snap-stabilizing PIF: –The system works as expected since its first wave.

5. Snap-stabilizing PIF and Useless Computations5 Context v State model (local shared memory model). v Related works: –Self-stabilizing solutions: u Cournier et al, 2001 –Snap-stabilizing solutions: u Cournier et al, 2002 u Blin et al, 2003 Assumption: every continuously enabled processor eventually executes an action Our snap-stabilizing PIF performs each PIF wave in a bounded number of steps

6. Snap-stabilizing PIF and Useless Computations6 PIF from a non-faulty configuration r

7. Snap-stabilizing PIF and Useless Computations7 And from an arbitrary configuration? r

8. Snap-stabilizing PIF and Useless Computations8 Problem: when performing the feedback? r r The processor must wait a correction

9. Snap-stabilizing PIF and Useless Computations9 Solution: a Question Mechanism [Blin et al, 2003] Q Reset Wait Only the root can deliver an answer and this answer is propagated into the tree of the root only Wait

10. Snap-stabilizing PIF and Useless Computations10 Answer: first case r Ok Wait Ok Wait

11. Snap-stabilizing PIF and Useless Computations11 Answer: second case r Ok The processor waits the correction Wait

12. Snap-stabilizing PIF and Useless Computations12 Erasing the abnormal PIFs: PIF on PIF Two kinds of abnormal PIFs: v PIFs rooted at p such that p ≠r –p is called « abnormal root » v Cycles –Level variables  at least one processor p detects that it is in the cycle –p is also considered as an « abnormal root »

13. Snap-stabilizing PIF and Useless Computations13 Erasing the abnormal PIFs: PIF on PIF v p is an abnormal root  p broadcasts a value that paralyses the abnormal PIF from p. v After p receives an acknowledgement from all its children, the abnormal PIF is erased from p.

14. Snap-stabilizing PIF and Useless Computations14 Problem: cleaning the PIF from r (the initiator) Two classical ways : v From the leaves to the root v From the root to the leaves PB: cleaning in one phase

15. Snap-stabilizing PIF and Useless Computations15 Problem of the cleaning in one phase r Receive the message for the 2 nd time

16. Snap-stabilizing PIF and Useless Computations16 Problem of the cleaning in one phase r...

17. Snap-stabilizing PIF and Useless Computations17 Solution: cleaning in two phases r

18. Snap-stabilizing PIF and Useless Computations18 Complexity Issues v Delay: –O(N) rounds. –O(∆ X N 3 ) steps. v A PIF wave: –O(N) rounds. –O(∆ X N 3 ) steps. v Memory requirement: –O(∆ X N) states per processor.

19. Snap-stabilizing PIF and Useless Computations19 Future work Transformer [Cournier et al, 2003]: v Based on the PIF and its applications. v Solutions with unbounded step complexities. Using our PIF to design a transformer that provides snap-stabilizing solutions with bounded step complexities.

20. Snap-stabilizing PIF and Useless Computations20 Useless Computations For any processor: v The number of corrupted reception is bounded by N. v The number of corrupted acknowledgement is bounded by 2.

21. Snap-stabilizing PIF and Useless Computations21 Thank you!