Near-Optimal Self-stabilising Counting and Firing Squads

Abstract

Consider a fully-connected synchronous distributed system consisting of n nodes, where up to f nodes may be faulty and every node starts in an arbitrary initial state. In the synchronous C-counting problem, all nodes need to eventually agree on a counter that is increased by one modulo C in each round for given C > 1. In the self-stabilising firing squad problem, the task is to eventually guarantee that all non-faulty nodes have simultaneous responses to external inputs: if a subset of the correct nodes receive an external "go" signal as input, then all correct nodes should agree on a round (in the not-too-distant future) in which to jointly output a "fire" signal. Moreover, no node should generate a "fire" signal without some correct node having previously received a "go" signal as input. We present a framework reducing both tasks to binary consensus at very small cost. For example, we obtain a deterministic algorithm for self-stabilising Byzantine firing squads with optimal resilience f < n/3, asymptotically optimal stabilisation and response time O( f ), and message size O(log f ). As our framework does not restrict the type of consensus routines used, we also obtain efficient randomised solutions.

Self-Stabilising Byzantine Clock Synchronisation Is Almost as Easy as Consensus

“…In contrast, we devise a new recursive scheme that allows us to (1) deterministically generate resynchronisation pulses in Θ(f ) time and (2) probabilistically generate resynchronisation pulses in o(f ) time. To construct algorithms that generate resynchronisation pulses, we employ resilience boosting and ltering techniques inspired by our recent line of work on digital clock synchronisation in the synchronous model [24,26]. One of its main motivations was to gain a be er understanding of the linear time/communication complexity barrier that research on pulse synchronisation ran into, without being distracted by the additional timing uncertainties due to communication delay and clock dri .…”

“…Any synchronous consensus routine can be converted into a silent consensus routine essentially for free. In our prior work [24], we showed that there exists a simple transformation that induces only an overhead of two rounds while keeping all other properties of the algorithm the same. eorem 5 ([24]).…”

“…Our pulse ltering scheme follows a similar idea as our recent construction of synchronous counting algorithms [24]. However, considerable care is needed to translate the approach from the (much simpler) synchronous round-based model to the bounded-delay model with clock dri .…”

“…We remark that the construction from [24] that generates silent consensus routines out of regular ones also applies to randomised algorithms (as produced by Lemma 30), that is, we can obtain suitable randomised silent consensus routines to be used in our framework.…”

“…Consequently, correct nodes stay in state until either the next pulse or T max,h expires. e former occurs at all correct nodes in V h no earlier than time p h,2 > p h,1 + T a + σ + 2d > p h,1 + T vote + σ + 2d by Constraint(19) and Constraint(24), and each pulse is received before time p h,2 +σ+d < p h,1 +Φ + h +T vote by Constraint(16) and Constraint(17). us, each correct node stay in state until time p h,2 with T max,h expiring no earlier than time p h,2 + σ + d. Consequently, we can repeat the above reasoning inductively to complete the proof.We de ne the following time bound for each h ∈ {0, 1}:6.5 Proof of Lemma 21Lemma 21 (Grouping lemma for validator machine).…”