1. Robust Stabilizing Leader Election
Carole Delporte-Gallet (LIAFA)
Stéphane Devismes (CNRS, LRI)
Hugues Fauconnier (LIAFA)
LIAFA

2. A convergence property
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.
A convergence property +
A closure property
November 14-16, 2007
SSS'07, Paris

3. Self- vs. Pseudo- Stabilization
Same Convergence Property
Illegitimate States
Legitimate States
November 14-16, 2007
SSS'07, Paris

4. Self- vs. Pseudo- Stabilization
Strong Closure vs. Ultimate Closure
Illegitimate States
Legitimate States
November 14-16, 2007
SSS'07, Paris

5. Self- vs. Pseudo- Stabilization
Strong Closure vs. Ultimate Closure
Illegitimate States
Legitimate States
November 14-16, 2007
SSS'07, Paris

6. In partial synchronous model ?
Robust Stabilization
Gopal and Perry, PODC’93
Beauquier and Kekkonen-Moneta, JSS’97
Anagnostou and Hadzilacos, WDAG’93
In partial synchronous model ?
Les deux premiers: systèmes synchrones
Le troisième: résultat d’impossibilité dans les réseaux asynchrones
November 14-16, 2007
SSS'07, Paris

7. weak reliability and synchrony assumptions
Leader Election
Robust Stabilizing Leader Election with:
weak reliability and synchrony assumptions
November 14-16, 2007
SSS'07, Paris

8. Model Network: fully-connected n Processes (numbered from 1 to n):
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
Two links between each pair of processes
November 14-16, 2007
SSS'07, Paris

9. Communication-Efficiency
[Larrea, Fernandez, and Arevalo, 2000]:
« An algorithm is communication-efficient if it eventually only uses n - 1 unidirectional links »
November 14-16, 2007
SSS'07, Paris

10. Self-Stabilizing Leader Election in a full timely network?
Yes + communication-efficiently
November 14-16, 2007
SSS'07, Paris

11. Principle of the algorithm
A process p periodically sends ALIVE to every other if Leader = p
Alive,1
Leader=1 4 3 2 1
Leader=1
Alive,1
Alive,1
Alive,2
Alive,2
Leader=2
Leader=2
Alive,2
November 14-16, 2007
SSS'07, Paris

12. Principle of the algorithm
When a process p such that Leader = p receives ALIVE from q, then Leader := q if q < p
Alive,1
Leader=1 4 3 2 1
Leader=1
Alive,1
Alive,1
Alive,2
Alive,2
Leader=2
Leader=1
Leader=2
Alive,2
November 14-16, 2007
SSS'07, Paris

13. Principle of the algorithm
Any process q such that Leader <> q always chooses as leader the process from which it receives ALIVE the most recently
Alive,1
Leader=1 4 3 2 1
Leader=1
Alive,1
Alive,1
Leader=2
Leader=1
Leader=1
November 14-16, 2007
SSS'07, Paris

14. Principle of the algorithm
On Time out, a process p sets Leader to p
Alive,1
Leader=1
Leader=3 4 3 2 1
Leader=1
Alive,1
Alive,1
Alive,2
Alive,2
Leader=2
Leader=2
Leader=4
Alive,2
November 14-16, 2007
SSS'07, Paris

15. Communication-Efficient Self-Stabilizing Leader Election in a system where at most one link is asynchronous? No
Is-it possible a weaker system ?
November 14-16, 2007
SSS'07, Paris

16. The process chooses another leader
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
Claim: not obligatorily from the leader
The process chooses another leader
November 14-16, 2007
SSS'07, Paris

17. Self-Stabilizing (non communication-efficient) Leader Election in a system where some links are asynchronous?
Yes
November 14-16, 2007
SSS'07, Paris

18. 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
November 14-16, 2007
SSS'07, Paris

19. 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 p periodically sends ALIVE,p to every other process
Any ALIVE,p message is repeated n - 1 times
(any other process periodically receives such a message)
November 14-16, 2007
SSS'07, Paris

20. Self-Stabilizing Leader Election in a system without timely routing overlay ?No
November 14-16, 2007
SSS'07, Paris

21. Pseudo-Stabilizing Leader Election in a system where Self-Stabilizing Leader Election is not possible ?
Yes + communication-efficiently
In a system having a source and fair links
(adaptation of an algorithm of [Aguilera et al, PODC’03])
November 14-16, 2007
SSS'07, Paris

22. Algorithm for systems with Source + fair links
A process p periodically sends ALIVE to every other if Leader = p
Each process stores in Active its ID + 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 2 1
Source
Alive,1
<1,2>
<1>
<1,2>
<2>
Alive,2
November 14-16, 2007
SSS'07, Paris

23. Systems with Source + fair links Accusation Counter
p stores in Counter[p] how many times it was suspected to be crashed
When p 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 1 2 3
Source 3 2
<3>
<1,3>
3,C=2
3,C=2
(we use IDs to break ties)
Accuse
1,C=1 1
<1,3> 4
<2>
<2,3>
1,C=1
November 14-16, 2007
SSS'07, Paris

24. Phase Counter 1 2 3 Source 2 <1,3> <3> 1 <1,3> 4
Each process maintains in Phase[p] the number of times it looses the leadership
p periodically 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] 1 2 3
Source 2
<1,3>
<3>
Ph=3
Ph=4 (previously 3)
3,C=2
3,C=2
1,C=1
Accuse,3
Ph=1 1
<1,3> 4
<2,3>
<2>
Ph=2
1,C=1
November 14-16, 2007
SSS'07, Paris

25. Communication-Efficient Pseudo-Stabilizing Leader Election in a system having only a source?
No, but a non communication-efficient pseudo-stabilizing leader election can be done (techniques similar to those used in the algorithm of [Aguilera et al, PODC’03])
November 14-16, 2007
SSS'07, Paris

26. Result Summary ce-SS SS ce-PS PS Full-Timely Yes Bi-source No
Timely routing ?
Source + fair links
Source
Totally asynchronous
November 14-16, 2007
SSS'07, Paris

27. Perspectives
Communication-efficient pseudo-stabilizing leader election in a system with timely routing overlay
Extend these results to other topologies and models
Robust stabilizing decision problems ?
Exemple de modèle: only a finite number of processes crashed
November 14-16, 2007
SSS'07, Paris

28. November 14-16, 2007
SSS'07, Paris