From Self- to Snap- Stabilization

1
From Self- to Snap- Stabilization
SSS2006, November 17-19, Dallas (USA)

• Alain Cournier, Stéphane Devismes, and Vincent
  Villain

2
Introduction

• Self-stabilizing protocols ? Snap-stabilizing
  protocols
• Arbitrary rooted network
• State model
• Local shared memory
• Daemon weakly fair/unfair

3
Related Work

• Transformer Cournier et al, 2003 in the state
  model
• Non fault-tolerant ? Snap-stabilizing
• Use Snapshots to regulary test if the system is
  in a normal configuration
• Drawbacks
• Define a predicat that caracterises the normal
  configuration
• The number of snapshots is unboundable
• Consequences
• the overcost of the transformer is difficult to
  evaluate
• scheduling assumption (at most a weakly fair
  deamon)

4
Assumptions

• The input protocol is
• Self-stabilizing
• Single-initiator wave protocol (the root is the
  initiator)
• Decision actions occur at the initiator only
• Example Token Circulation, PIF, Spanning tree
  construction (DFS or BFS)

5
Self- vs Snap- Stabilizing Wave Protocols

• A self-stabilizing wave protocol converges to a
  specified behavior in a finite time.
• N is finite but generally unbounded

F(.)
6
Self- vs Snap- Stabilizing Wave Protocols

• Since its first starting action (the real start
  of the protocol), a snap-stabilizing wave
  protocol works according to its specification.
• Consequence a snap-stabilizing wave protocol do
  not require to be repeated.

F(.)
7
More precisely
A snap-stabilizing wave protocol for a task T
verifies
T is executed as expected
Configurations
Time
Decision
Request
Starting Action
8
Our solution

• Let P be self-stabilizing wave protocol for a
  task T.
• We compose P with Reset protocol as follows

P executes T
One Reset
Configurations
Time
Decision
Request
Starting Action
9
Our solution

• Problem when a computation of T is requested
• The Reset must start in a finite time
• But without aborting a previous initiated
  computation of T
• Solution we use a boolean Endr
• Endr True at the decision (as P is
  self-stabilizing, P eventually decides)
• While Endr True, P cannot start a computation
  of T
• Endr True causes a Reset of the P Variables
• At the end of the Reset, Endr false

10
Snap-stabilizing Reset

• Using a snap-stabilizing PIF protocol
• 2 phases broadcast and Feedback
• The processors abort the computation of T when
  receiving the broadcast phase
• The reset is performed during the feedback phase
• The snap-stabilizing PIF of Cournier et al,
  2006
• Bounded step complexity (unfair deamon)

This implies that the transformer works at least
with the same deamon that the initial protocol
11
Case Study DFTC of Huang and Chen, 1993
R
Correct behavior
12
Case Study DFTC of Huang and Chen, 1993

• Starting for an abnormal initial configuration
• Abnormal successor paths
• Correction using a third color ERROR, the
  abnormal successor paths are paralysed before to
  be removed.
• Problem

R
Can never move if the deamon is unfair
13
With the transformer

• At least a weakly fair daemon

R
Endr
Decision in a finite number of steps
Reset in a finite number of steps
Token circulation in a finite number of steps
14
Complexity ?

• Stabilization time of Huang and Chen, 1993
• ?(n?D) rounds

R
15
Complexity with the transformer

• Decision O(N) rounds
• Reset O(N) rounds Cournier et al, 2006
• Token circulation O(N) rounds

R
Endr
16
Conclusion

• Simple
• Low memory overcost memory requirement of the
  reset protocol (O(log N) bits)
• At least the same scheduling assumption
• In some cases
• Huang and Chen, 1993, Johnen and Beauquier,
  1995, Datta et al, 1998
• Better scheduling assumption (Weakly Fair ?
  Unfair)
• Better time complexity (?(n?D) rounds ? O(N)
  rounds)

17
Perspective

• Apply a similar technique to transform Non
  Fault-Tolerant Wave Protocols into
  Snap-Stabilizing Wave Protocols
• (done).
• Multi-initiators

18
Thank you!