Trade-off between Time, Space, and Workload: the case of the Self-stabilizing Unison

TL;DR

This paper introduces a novel self-stabilizing algorithm for the asynchronous unison problem in anonymous networks, optimizing time, space, and workload, and enabling efficient simulation of synchronous algorithms in asynchronous settings.

Contribution

It presents the first self-stabilizing unison algorithm for anonymous networks with optimal stabilization time and bounded memory, improving leader election and spanning tree construction.

Findings

Achieves stabilization in at most 2D-2 rounds.

Uses only O(log B) bits per node.

Enables efficient simulation of synchronous algorithms.

Abstract

We present a self-stabilizing algorithm for the (asynchronous) unison problem which achieves an efficient trade-off between time, workload, and space in a weak model. Precisely, our algorithm is defined in the atomic-state model and works in anonymous networks in which even local ports are unlabeled. It makes no assumption on the daemon and thus stabilizes under the weakest one: the distributed unfair daemon. In a $n$-node network of diameter $D$ and assuming a period $B \geq 2D+2$, our algorithm only requires $O(\log B)$ bits per node to achieve full polynomiality as it stabilizes in at most $2D-2$ rounds and $O(\min(n^2B, n^3))$ moves. In particular and to the best of our knowledge, it is the first self-stabilizing unison for arbitrary anonymous networks achieving an asymptotically optimal stabilization time in rounds using a bounded memory at each node. Finally, we show that our solution allows to efficiently simulate synchronous self-stabilizing algorithms in an asynchronous environment. This provides a new state-of-the-art algorithm solving both the leader election and the spanning tree construction problem in any identified connected network which, to the best of our knowledge, beat all existing solutions of the literature.