Introduction%20to%20Self-Stabilization

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Introduction to Self-Stabilization

• Stéphane Devismes (CNRS, LRI)

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Example of Self-Stabilizing System

• Dijkstras Token Ring

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Model

• Locally Shared Memory
• Guarded Action
• Action
• Executed only if its guard is true (enabled)
• The execution is asynchronous but each step is
  atomic

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Topology Rooted Oriented Ring
P0
P1
P5
P2
P4
P3
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Algorithm (K 7)
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Transient Faults (undefinitive and rare)
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1
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The system retreives by itself a correct
behavior Self-stabilization
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Self-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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Self-Stabilization
Closure
Illegitimate States
Legitimate States
Convergence
System States
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Advantages

• Fault-Tolerance
• Initialization
• Dynamic Topology

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Disavantages

• Initial inconsistencies (stabilization time)
• Overcost
• No detection of stabilization

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Around Self-Stabilization

• Probabilistic Self-Stabilization
• Robust Stabilization
• Weak-Stabilization
• Pseudo-Stabilization
• Snap-Stabilization
• Fault-Containment

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Robust Stabilization

• The system stabilizes even if some processes crash

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Pseudo-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.
• Self ? Pseudo ?

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Self- vs Pseudo-

• Specification (i,i,i,),(j,j,j,)

i
r
i
j
j
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Robust Stabilizing Leader Election(SSS07)

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

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SSS07 (WRAS)

• 9th International Symposium on Stabilization,
  Safety, and Security of Distributed Systems
• 14-16 November 2007, Paris, France
• http//sss07.lri.fr/

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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)
• timely
• can crashed (an arbitrary number of processes may
  crash)
• Variables initially arbitrary assigned
• Links
• Unidirectional
• 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 p

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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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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

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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

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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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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 source and fair
  links (adaptation of an algorithm of Aguilera et
  al, PODC93)

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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 an Active set 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

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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,ph with ph Phasep

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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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Perspectives

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