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Introduction to Self-Stabilization

Introduction to Self-Stabilization. Stéphane Devismes. Self-Stabilization [Dijkstra, 1974]. Example: Dijkstra’s Token Ring. 0. 1. 2. 1. 0. 1. 0. 1. 0. 0. 1. Starting from an arbitrary state. 5. 4. 0. 2. 5. 1. 0. 4. 0. 5. 5. Definition: Closure + Convergence. Closure.

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Introduction to Self-Stabilization

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  1. Introduction to Self-Stabilization StéphaneDevismes

  2. Self-Stabilization [Dijkstra, 1974] • Example: Dijkstra’s Token Ring 0 1 2 1 0 1 0 1 0 0 1

  3. Starting from an arbitrary state 5 4 0 2 5 1 0 4 0 5 5

  4. Definition: Closure + Convergence Closure Illegitimate states Legitimate States Convergence States of the system

  5. Why Self-Stabilization? Advantages Drawbacks

  6. Protocols for: • Resources Allocation (Mutual Exclusion) • Broadcast • Routing • Overlay (Spanning trees, Routing table) • …

  7. Around Self-Stabilization (1/2) • Weaker Properties: • K-Stabilization (no more than K faults) • Weak-Stabilization (possible convergence) • Probabilistic Stabilization (probabilistic convergence) • Pseudo-Stabilization • Aim: circumvent impossibility results • Example: alternated bit protocol

  8. Pseudo-Stabilization ? • Self-Stabilization [Dijkstra, 1974]: Starting from any configuration, a self-stabilizing system reaches in a finite time a configuration c such that any suffix starting from c satisfies the intended specification. • Pseudo-Stabilization [Burns, Gouda, and Miller, 1993]: Starting from any configuration, any execution of a pseudo-stabilizing system has a non-empty suffix that satisfies the intended specification.

  9. Self- vs. Pseudo- Stabilization Strong Closure vs. Ultimate Closure Illegitimate States Legitimate States

  10. Self- vs. Pseudo- Stabilization • Example: Leader Election • Self-Stabilizing Leader Election: • Eventually there is a unique leader that cannot change • Pseudo-Stabilizing Leader Election: • We never have the guarantee that the leader no more changes but eventually it no more change • Remark: no stabilization time in pseudo-stabilization

  11. Around Self-Stabilization (2/2) • Stronger Properties: • Fault-containment (Quick stabilization when there are few faults) • Snap-Stabilization (Safety for the tasks started after the faults) • Byzantine-Tolerant Stabilization • Fault-Tolerant Stabilization(Stabilization despite crashes) • Aim: circumvent the drawbacks

  12. LIAFA Fault-Tolerant Stabilizing Leader Election CaroleDelporte-Gallet(LIAFA) StéphaneDevismes(CNRS, LRI) HuguesFauconnier(LIAFA)

  13. Fault-Tolerant Stabilization • Gopal and Perry, PODC’93 • Beauquier and Kekkonen-Moneta, JSS’97 • Anagnostou and Hadzilacos, WDAG’93 In partial synchronous model ?

  14. Leader Election Fault-Tolerant Stabilizing Leader Election with: weak reliability and synchrony assumptions

  15. Model • Network: fully-connected • n Processes: • timely • may crash (an arbitrary number of processes may crash) • Variables: initially arbitrary assigned • Links: • Unidirectional • Initially not necessarily empty • No order on the message deliverance • Variable reliability and timeliness assumptions

  16. Communication-Efficiency [Larrea, Fernandez, and Arevalo, 2000]: « An algorithm is communication-efficient if it eventually only uses n - 1 unidirectional links »

  17. Self-Stabilizing Leader Electionin a full timely network? Yes + communication-efficiency

  18. Principles of the algorithm • A process p periodically sends ALIVE to every other if Leader = p Alive,1 1 4 Leader=1 Alive,1 Alive,1 Alive,2 Alive,2 3 2 Leader=2 Leader=2 Alive,2

  19. 1 4 3 2 Principles of the algorithm • When a process p such that Leader = p receives ALIVE from q, then • Leader := qif q < p Alive,1 Leader=1 4 Alive,1 Alive,1 Alive,2 Alive,2 Leader=2 Leader=2 Leader=1 Alive,2

  20. 1 4 3 2 Principles of the algorithm • Any process q such that Leader ≠ q always chooses as leader the process from which it receives ALIVEthe most recently Alive,1 Leader=1 4 Alive,1 Alive,1 Leader=2 Leader=1 Leader=1

  21. 1 4 3 2 Principles of the algorithm • On Time out, a process p sets Leader to p Alive,1 Leader=3 Leader=1 4 Alive,1 Alive,1 Alive,2 Alive,2 Leader=2 Leader=4 Leader=2 Alive,2

  22. Communication-EfficientSelf-StabilizingLeader Electionin a system where at most one link is asynchronous? No

  23. Impossibility of Communication-Efficiency in a system with at most one asynchronous link • Claim: Any process p such that Leader ≠ p must periodically receive messages within a bounded time otherwise it chooses another leader The process chooses another leader

  24. Self-Stabilizing (non communication-efficient)Leader Electionin a system where some links are asynchronous? Yes

  25. Self-Stabilizing Leader Election in a system with a timely routing overlay • For each pair of alive processes (p,q), there exists at least two paths of timely links: • From p to q • From q to p

  26. Principle of the algorithm • Each process computes the set of alive processes and chooses as leader the smallest process of this set • To compute the set: • Each process pperiodically sends ALIVE,p to every other process • Any ALIVE,p message is repeated n- 1 times (any other process periodically receives such a message)

  27. Self-Stabilizing Leader Electionin a system without timely routing overlay ? No

  28. Pseudo-Stabilizing Leader Electionin a system where Self-Stabilizing Leader Election is not possible ? • Yes + communication-efficiently • In a system having a source and fair links

  29. Source 1 2 Algorithm for systems with Source + fair links • A process pperiodically sends ALIVE to every other if Leader = p • Each process stores in Active itsID+ the IDs of each process from which it recently receives ALIVE • Each process chooses its leader among the processes in its Active set • Problem: we cannot use the IDs to choose a leader Alive,1 <1,2> <1> <1,2> <2> Alive,2

  30. Source 3 1 2 Accusation Counter • p stores in Counter[p] how many times it was suspected to be crashed • When a process suspects its leader: • it sends an ACCUSATION to LEADER, and • chooses as new leader the process in Active with the smallest accusation counter • p periodically sends ALIVE,Counter[p] to every other if Leader = p • Problem: the accusation counter of the source can increase infinitely often 2 <3> 3 <1,3> 3,C=2 3,C=2 Accuse 1,C=1 4 <2> <2,3> 1 <1,3> 1,C=1

  31. Source 3 1 2 Phase Counter • Each process maintains in Phase[p] the number of times it looses the leadership • pperiodically sends ALIVE,Counter[p],Phase[p] to every other if Leader = p • p increments Counter[p] only when receiving ACCUSATION,ph with ph = Phase[p] 2 <1,3> <3> Ph=4 (previously 3) Ph=3 3,C=2 3,C=2 1,C=1 Accuse,3 1 <1,3> 4 <2,3> Ph=1 <2> Ph=2 1,C=1

  32. Communication-Efficient Pseudo-Stabilizing Leader Electionin a system having only a source? No, but a non communication-efficient pseudo-stabilizing leader election can be done

  33. Result Summary

  34. Perspectives • Communication-efficient FTPS leader election in a system with timely routing overlay • Extend these results to other topologies and models • Fault-tolerant stabilizing decision problems ?

  35. Thank You!

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