Robust Stabilizing Leader Election (SSS

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Robust Stabilizing Leader Election(SSS07)

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

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Context

• Two Worlds
• Robust Algorithms
• Withstand crash failures
• Stabilizing Algorithms
• Tolerate transient failures (temporary and rare)

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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.

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Stabilization

• 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.

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Self- vs Pseudo-

• Specification (i,i,i,),(j,j,j,)

i
r
i
j
j
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Related Works on Robust Stabilization

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

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Model

• Network fully-connected
• n Processes (numbered from 1 to n) initially
  crashed (an arbitrary number of processes may
  crash) or timely forever
• Variables initially arbitrary assigned
• Links
• Initially not necessarily empty
• No order on the message delivrance
• 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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Can we implement Self-Stabilizing Leader Election
in a full synchronous network?
Yes, it can be communication-efficiently
implemented
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Principle of the algorithm

1. A process p periodically sends ALIVE to every
   other if Leader p
2. Any process q such that Leader ltgt q always
   chooses as leader the process from which it
   receives ALIVE the most recently
3. When a process p such that Leader p receives
   ALIVE from q, then Leader q if q lt p
4. On Time out, a process p sets Leader to q

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Can we implement Communication-Efficient
Self-Stabilizing Leader Election in a system
where at most one link is asynchronous?
No
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Sketch of Proof

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

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Can we implement (non communication efficient)
Self-Stabilizing 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 processor (p,q), there
  exists at least two paths of timely links
• From p to q
• From q to p
• Particular case timely bi-source, i.e.,
• A timely process having all its (outgoing and
  ingoing) links that are timely

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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)
• REMARK for a system with a timely bi-source, we
  just have to repeat the messages only once.

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Can we implement Self-Stabilizing Leader Election
in a system without timely routing overlay ?
No
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Can we implement a Communication-Efficient
Pseudo-Stabilizing Leader Election in a system
where Communication-Efficient Self-Stabilizing
Leader Election is not possible ?

• Yes
• In a system having a timely bi-source (Algorithm
  of Aguilera et al, DISC01)
• In a system having a timely source and fair
  links (adaptation of an algorithm of Aguilera et
  al, PODC93)

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Principle of the algorithm of Aguilera et al,
PODC93

• A process p periodically sends ALIVE to every
  other if Leader p
• Each process store in an Active set the IDs of
  each process from which it recently receives
  ALIVE
• Each process choose its leader among the
  processes in its Active set
• Problem we cannot use the IDs to choose a leader

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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 its
  Active set with the smallest accusation counter
  (we use IDs to break ties)
• p periodically sends ALIVE,Counterp to every
  other if Leader p
• Problem assuming that LEADERs, the source s can
  volontary stop sending ALIVE

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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,q with q Counterp

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Can we implement a Communication-Efficient
Pseudo-Stabilizing Leader Election in a system
having only a timely 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, PODC93)
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Result Summary
ce-SS SS ce-PS PS
Synchronous Yes Yes Yes Yes
Timely bi-source No Yes Yes Yes
Timely routing No Yes ? Yes
Timely source fair links No No Yes Yes
Timely source No No No Yes
Totally asynchronous No No No No
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Concluding Remarks

• Our algorithms also work with eventual timely
  assumptions
• Consider initial crash failures is not a
  restriction
• In fix-point problem, the gap between robustness
  and stabilizing robustness is not really
  significant

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Perspectives

• Communication-efficient leader election in a
  system with timely routing
• Extend these results to other topologies and
  models
• Robust stabilizing decision problems ?