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Snap-Stabilizing PIF and Useless Computations

ICPADS’2006, July 12-15 2006, Minneapolis (USA). Snap-Stabilizing PIF and Useless Computations. Alain Cournier, Stéphane Devismes , and Vincent Villain. PIF scheme. PIF = Propagation of informations with feedback Any processor can be the initiator of a PIF wave. A PIF wave :

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Snap-Stabilizing PIF and Useless Computations

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

  2. PIF scheme • PIF = Propagation of informations with feedback • Any processor can be the initiator of a PIF wave. • 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. Snap-stabilizing PIF and Useless Computations

  3. 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 specification. [Bui et al, 1999] • Application: these systems tolerates transient failure Snap-stabilizing PIF and Useless Computations

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  18. Complexity Issues • Delay: • O(N) rounds. • O(∆ X N3) steps. • A PIF wave: • O(N) rounds. • O(∆ X N3) steps. • Memory requirement: • O(∆ X N) states per processor. Snap-stabilizing PIF and Useless Computations

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

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

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

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