A Snap-Stabilizing DFS with a Lower Space Requirement SSS’2005, October 26-27 2005, Barcelona (Spain) A Snap-Stabilizing DFS with a Lower Space Requirement Alain Cournier, Stéphane Devismes, and Vincent Villain

Snap-Stabilizing System? A particular class of self-stabilizing systems. (A self-stabilizing system, regardless of the initial state of the processors, is guaranteed to converge to the intended behavior in finite time. (Dijkstra 1974)) A snap-stabilizing system, regardless of the initial state of the processors, always behaves according to its specifications. (Bui et al, 1999) A Snap-Stabilizing DFS with a Lower Space Requirement

A Snap-Stabilizing DFS with a Lower Space Requirement Model R Arbitrary rooted network State model Local shared memory Daemon: weakly fair/unfair A Snap-Stabilizing DFS with a Lower Space Requirement

O(NN) states (Ids List) Related Works Self-stabilizing area: Huang and Chen (Distributed Computing, 1993) Johnen and Beauquier (WSS, 1995) Johnen et al (WDAG, 1997) Datta et al (Distributed Computing, 2000) Snap-stabilizing area: Cournier et al, Transformer (ICDCS, 2003) Cournier et al (OPODIS, 2004) Weakly fair Daemon Unfair Daemon O(NN) states (Ids List) A Snap-Stabilizing DFS with a Lower Space Requirement

A Snap-Stabilizing DFS with a Lower Space Requirement Write a snap-stabilizing unfair DFS protocol with a lower memory requirement (no Ids) A Snap-Stabilizing DFS with a Lower Space Requirement

DFS in a non-faulty configuration A Snap-Stabilizing DFS with a Lower Space Requirement

Problem of the cleaning by the leaves Visited for the 2nd time A Snap-Stabilizing DFS with a Lower Space Requirement

Problem of the cleaning by the leaves ... ... A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: cleaning at the end of the traversal only A Snap-Stabilizing DFS with a Lower Space Requirement

Insuring the visit of all the processors A Snap-Stabilizing DFS with a Lower Space Requirement

Insuring the visit of all the processors A Snap-Stabilizing DFS with a Lower Space Requirement

Insuring the visit of all the processors Processors never visited r Not Snap-stabilizing! A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: a Question Mecanism [Blin et al, An Improved Snap-Stabilizing PIF Algorithm, 2003] Can I switch to the red color? Only the root can deliver an answer and this answer is propagated in the spanning tree of the root only A Snap-Stabilizing DFS with a Lower Space Requirement

Insuring the visit of all the processors Ok Ok Ok Ok r Ok Ok Ok A Snap-Stabilizing DFS with a Lower Space Requirement

Insuring the visit of all the processors Ok Ok Ok The yellow processor waits Until the abnormal traversal disappears by Error Correction A Snap-Stabilizing DFS with a Lower Space Requirement

A Snap-Stabilizing DFS with a Lower Space Requirement Error Correction 1 7 2 8 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 1 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 2 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 3 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 6 R 5 4 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 6 R 5 A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism 6 R A Snap-Stabilizing DFS with a Lower Space Requirement

Solution: Paralyzing Mechanism A Snap-Stabilizing DFS with a Lower Space Requirement

A Snap-Stabilizing DFS with a Lower Space Requirement Conclusion Snap-Stabilizing Arbitrary Rooted Networks Unfair Daemon A Snap-Stabilizing DFS with a Lower Space Requirement

Conclusion: SSS’05 vs OPODIS’04 Memory Requirement O(NN) O(∆²× N) Ids Yes No Stabilization time Delay (rounds) O(N) O(N²) Delay (steps) O(N3) Execution (rounds) Execution (steps) In fact, our protocol stabilizes in 0 times but the complexity of the delay to start a token circulation as well as the complexity for execute a complete token circulation are worst than the result obtained with the protocol of Cournier and al. A Snap-Stabilizing DFS with a Lower Space Requirement

Perspective Finding an efficient solution in both time and space complexities A Snap-Stabilizing DFS with a Lower Space Requirement

A Snap-Stabilizing DFS with a Lower Space Requirement Thank You! A Snap-Stabilizing DFS with a Lower Space Requirement