Compact deterministic self-stabilizing leader election on a ring: the exponential advantage of being talkative

Abstract

This paper focuses on compact deterministic self-stabilizing solutions for the leader election problem. When the protocol is required to be silent (i.e., when communication content remains fixed from some point in time during any execution), there exists a lower bound of Ω(log n) bits of memory per node participating to the leader election (where n denotes the number of nodes in the system). This lower bound holds even in rings. We present a new deterministic (non-silent) self-stabilizing protocol for n-node rings that uses only O(log log n) memory bits per node, and stabilizes in O(n log 2 n) rounds. Our protocol has several attractive features that make it suitable for practical purposes. First, the communication model fits with the model used by existing compilers for real networks. Second, the size of the ring (or any upper bound on this size) needs not to be known by any node. Third, the node identifiers can be of various sizes. Finally, no synchrony assumption, besides a weakly fair scheduler, is assumed. Therefore, our result shows that, perhaps surprisingly, trading silence for exponential improvement in term of memory space does not come at a high cost regarding stabilization time or minimal assumptions.

Compact Self-Stabilizing Leader Election for General Networks

“…For this reasons, after a node v detects an error, our algorithm cleans v and all of its descendants. The cleaning process is achieved by Algorithm Freeze, already presented in previous works [12,13,10]. Algorithm Freeze is run in two cases: cycle detection (thanks to predicate ErCyclepvq, presented in Subsection 3.4), and impostor leader detection (thanks to predicate ErSTpvq, presented in Subsection 3.5).…”

“…There are a few self-stabilizing tree-construction algorithms that are not using explicit distance variables (see, e.g., [30,22,16]), but their space-complexity is Opn log nq bits [22,16] or Oplog n`∆q [30]. Using the general principle of distance variables with spacecomplexity below Θplog nq bits was attempted by Awerbuch et al [5], and Blin et al [12]. These papers distribute pieces of information about the distances to the leader among the nodes according to different mechanisms, enabling to store oplog nq bits per node.…”

“…An algorithm is silent if each of its executions reaches a point in time after which the states of nodes do not change. A non-silent algorithm is said to be talkative (see[13]). …”

A self-stabilizing memory efficient algorithm for the minimum diameter spanning tree under an omnipotent daemon

“…The first open question is to decide if our non silent self-stabilizing algorithm is optimal with respect to memory requirement. A recent work [10], which presents a non silent BFS-based leader election self-stabilizing algorithm requiring O(log log n) bits of memory per process, leads us to think that we can improve the memory requirement of our algorithm. This naturally opens another question about the optimality of a silent self-stabilizing algorithm for minimum diameter spanning tree construction.…”

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.…”