From Self- to Snap- Stabilization Alain Cournier, Stéphane Devismes, and Vincent Villain SSS’2006, November 17-19, Dallas (USA)

From Self- to Snap- Stabilization2 Introduction  Self-stabilizing protocols → Snap-stabilizing protocols v Arbitrary rooted network v State model –Local shared memory –Daemon: weakly fair/unfair

From Self- to Snap- Stabilization3 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 : u Define a predicat that caracterises the normal configuration u The number of snapshots is unboundable –Consequences : u the overcost of the transformer is difficult to evaluate u scheduling assumption (at most a weakly fair deamon)

From Self- to Snap- Stabilization4 Assumptions v The input protocol is: –Self-stabilizing –Single-initiator wave protocol (the root is the initiator) –Decision actions occur at the initiator only v Example: Token Circulation, PIF, Spanning tree construction (DFS or BFS)…

From Self- to Snap- Stabilization5 A self-stabilizing wave protocol converges to a specified behavior in a finite time. N is finite but generally unbounded Self- vs Snap- Stabilizing Wave Protocols 1.X ≠ F(X) F(.)2.X ≠ F(X) 3.X ≠ F(X) 4.X ≠ F(X) N.X F(X)

From Self- to Snap- Stabilization6 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. 1.X F(X) F(.)

From Self- to Snap- Stabilization7 More precisely Configurations Time RequestStarting Action A snap-stabilizing wave protocol for a task T verifies : Decision T is executed as expected

From Self- to Snap- Stabilization8 Our solution Let P’ be self-stabilizing wave protocol for a task T. We compose P’ with Reset protocol as follows : Configurations Time RequestStarting Action Decision P’ executes T One Reset

From Self- to Snap- Stabilization9 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 End r : –End r := True at the decision (as P’ is self-stabilizing, P’ eventually decides) –While End r = True, P’ cannot start a computation of T –End r := True causes a Reset of the P’ Variables –At the end of the Reset, End r := false

From Self- to Snap- Stabilization10 Snap-stabilizing Reset Using a snap-stabilizing PIF protocol : v 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 v 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

From Self- to Snap- Stabilization11 Case Study : DFTC of [Huang and Chen, 1993] R Correct behavior

From Self- to Snap- Stabilization12 Case Study : DFTC of [Huang and Chen, 1993] Starting for an abnormal initial configuration : v Abnormal successor paths –Correction : using a third color ERROR, the abnormal successor paths are paralysed before to be removed. v Problem : R Weakly Fair Deamon Can never move if the deamon is unfair

From Self- to Snap- Stabilization13 With the transformer … At least a weakly fair daemon R Unfair deamon End r Decision in a finite number of steps Reset in a finite number of steps Token circulation in a finite number of steps

From Self- to Snap- Stabilization14 Complexity ? Stabilization time of [Huang and Chen, 1993] :  (n  D) rounds R

From Self- to Snap- Stabilization15 Complexity with the transformer Decision : O(N) rounds Reset : O(N) rounds [Cournier et al, 2006] Token circulation : O(N) rounds R A correct Token circulation : O(N) rounds End r

From Self- to Snap- Stabilization16 Conclusion v Simple v Low memory overcost : memory requirement of the reset protocol (O(log N) bits) v At least the same scheduling assumption v 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)

From Self- to Snap- Stabilization17 Perspective v Apply a similar technique to transform Non Fault-Tolerant Wave Protocols into Snap-Stabilizing Wave Protocols (done). v Multi-initiators

From Self- to Snap- Stabilization18 Thank you!