Self-stabilizing Leader Election in Polynomial Steps

Abstract

We propose a silent self-stabilizing leader election algorithm for bidirectional connected identified networks of arbitrary topology. This algorithm is written in the locally shared memory model. It assumes the distributed unfair daemon, the most general scheduling hypothesis of the model. Our algorithm requires no global knowledge on the network (such as an upper bound on the diameter or the number of processes, for example). We show that its stabilization time is in Θ(n 3) steps in the worst case, where n is the number of processes. Its memory requirement is asymptotically optimal, i.e., Θ(log n) bits per processes. Its round complexity is of the same order of magnitude-i.e., Θ(n) rounds-as the best existing algorithm [10] designed with similar settings. To the best of our knowledge, this is the first self-stabilizing leader election algorithm for arbitrary identified networks that is proven to achieve a stabilization time polynomial in steps. By contrast, we show that the previous best existing algorithm designed with similar settings [10] stabilizes in a non polynomial number of steps in the worst case.

Acyclic Strategy for Silent Self-stabilization in Spanning Forests

“…To illustrate the versatility of our method, we review several existing works where our results apply.After the seminal work of Dijkstra, many self-stabilizing algorithms have been proposed to solve various tasks such as spanning tree constructions [2], token circulations [3], clock synchronization [4], propagation of information with feedbacks [5]. Those works consider a large taxonomy of topologies: rings [6,7], (directed) trees [5,8,9], planar graphs [10,11], arbitrary connected graphs [12,13], etc. Among those topologies, the class of directed (in-) trees (i.e., trees where one process is distinguished as the root and edges are oriented toward the root) is of particular interest.…”

“…In other words, these latter processes waste computation power and so energy. Such a situation should be therefore prevented, making the unfair daemon more desirable than the weakly fair one.There are many self-stabilizing algorithms proven under the distributed unfair daemon, e.g., [13,18,19,20,21]. However, analyses of the stabilization time in moves is rather unusual and this may be an important issue.…”

“…However, analyses of the stabilization time in moves is rather unusual and this may be an important issue. Indeed, recently, several self-stabilizing algorithms which work under a distributed unfair daemon have been shown to have an exponential stabilization time in moves in the worst case, e.g., the silent leader election algorithms from [19,20] (as shown in [13]), the Breadth-First Search (BFS) algorithm of Huang and Chen [22] (as shown in [23]), or the silent self-stabilizing algorithm for the shortest-path spanning tree of [21] (as…”