An O(D)-Round Snap-stabilizing Termination Detection in Arbitrary Rooted Networks
Résumé
In this paper, we investigate the question of (asymptotic) time optimality for snap-stabilizing wave algorithms. Snap-stabilization is a refinement of self-stabilization guaranteeing that after a period of transient faults, any system achieving it immediately resumes a correct behavior. We consider here the atomic-state model under the distributed weakly fair daemon in the context of rooted connected networks. In these settings, we propose to augment a silent BFS spanning tree construction with efficient snap-stabilizing termination detection. This approach yields an asynchronous snap-stabilizing wave algorithm that both constructs a BFS spanning tree and detects the termination of its construction in O(D) rounds and using O(log ∆ + log D) bits per process, where ∆ is the maximum degree of the network and D is any upper bound on D. As a result, and to the best of our knowledge, we obtain the first asynchronous asymptotically round-optimal snap-stabilizing wave algorithm of the literature. We then generalize our approach to augment any silent algorithm with snap-stabilizing termination detection. This leads to a snap-stabilizing wave algorithm that both executes the silent task and detects its termination in O(F + D) rounds, where F is the stabilization time of the input silent algorithm; thereby incurring an asymptotically optimal additive overhead of O(D) rounds. The memory overhead also remains low, namely O(log ∆ + log D) additional bits per process.
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