Robust Stabilizing Leader Election

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Robust Stabilizing Leader Election

• Carole Delporte-Gallet (LIAFA)
• Stéphane Devismes (CNRS, LRI)
• Hugues Fauconnier (LIAFA)

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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
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Self- vs. Pseudo- Stabilization
Same Convergence Property
Illegitimate States
Legitimate States
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Self- vs. Pseudo- Stabilization
Strong Closure vs. Ultimate Closure
Illegitimate States
Legitimate States
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Self- vs. Pseudo- Stabilization
Strong Closure vs. Ultimate Closure
Illegitimate States
Legitimate States
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Robust Stabilization

• Gopal and Perry, PODC93
• Beauquier and Kekkonen-Moneta, JSS97
• Anagnostou and Hadzilacos, WDAG93
• In partial synchronous model ?

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Leader Election

• Robust Stabilizing Leader Election with
• weak reliability and synchrony assumptions

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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

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Communication-Efficiency

• Larrea, Fernandez, and Arevalo, 2000
•  An algorithm is communication-efficient if it
  eventually only uses n - 1 unidirectional links 

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Self-Stabilizing Leader Election in a full
timely network?
Yes communication-efficiently
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Principle of the algorithm

• A process p periodically sends ALIVE to every
  other if Leader p

Alive,1
Leader1
Leader1
Alive,1
Alive,1
Alive,2
Alive,2
Leader2
Leader2
Alive,2
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Principle of the algorithm

• When a process p such that Leader p receives
  ALIVE from q, then Leader q if q lt p

Alive,1
Leader1
Leader1
Alive,1
Alive,1
Alive,2
Alive,2
Leader2
Leader2
Leader1
Alive,2
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Principle of the algorithm

• Any process q such that Leader ltgt q always
  chooses as leader the process from which it
  receives ALIVE the most recently

Alive,1
Leader1
Leader1
Alive,1
Alive,1
Leader2
Leader1
Leader1
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Principle of the algorithm

• On Time out, a process p sets Leader to p

Alive,1
Leader3
Leader1
Leader1
Alive,1
Alive,1
Alive,2
Alive,2
Leader2
Leader4
Leader2
Alive,2
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Communication-Efficient Self-StabilizingLeader
Election in a system where at most one link is
asynchronous?
No
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Impossibility of Communication-Efficiency in a
system with at most one asynchronous link

• Claim Any process p such that Leader ltgt p must
  periodically receive messages within a bounded
  time otherwise it chooses another leader

The process chooses another leader
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Self-Stabilizing (non communication-efficient)
Leader Election in a system where some links
are asynchronous?
Yes
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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

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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)

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Self-Stabilizing Leader Election in a system
without timely routing overlay ?
No
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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,
  PODC03)

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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

Alive,1
lt1,2gt
lt1gt
lt2gt
lt1,2gt
Alive,2
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Systems with Source fair links Accusation
Counter

• p stores in Counterp 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,Counterp to every
  other if Leader p
• Problem the accusation counter of the source can
  increase infinitely often

lt3gt
2
3
lt1,3gt
3,C2
3,C2
Accuse
1,C1
lt2,3gt
4
lt2gt
1
lt1,3gt
1,C1
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Phase Counter

• Each process maintains in Phasep the number of
  times it looses the leadership
• p periodically sends ALIVE,Counterp,Phasep
  to every other if Leader p
• p increments Counterp only when receiving
  ACCUSATION,ph with ph Phasep

lt3gt
2
lt1,3gt
Ph3
Ph4 (previously 3)
3,C2
3,C2
1,C1
Accuse,3
1
lt2,3gt
4
lt1,3gt
Ph1
Ph2
lt2gt
1,C1
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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, PODC03)
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Result Summary
ce-SS SS ce-PS PS
Full-Timely Yes Yes Yes Yes
Bi-source No Yes Yes Yes
Timely routing No Yes ? Yes
Source fair links No No Yes Yes
Source No No No Yes
Totally asynchronous No No No No
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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 ?

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