Near-optimal self-stabilising counting and firing squads

Consider a fully-connected synchronous distributed system of n nodes, where up to f nodes may be faulty and every node starts in an arbitrary initial state. In the synchronous counting problem, all nodes need to eventually agree on a counter that is increased by one modulo some C in each round. 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: we maintain the resilience of the underlying consensus routine, while the stabilisation time and message size are, up to constant factors, bounded by the sum of the cost of the consensus routine for f faults and recursively applying our scheme to f’<f/2 faults. 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, and it is straightforward to adapt our framework to allow for f<n/2 omission or f<n crash faults, respectively. Our results resolve various open questions on the two problems, most prominently whether (communication-efficient) self-stabilising Byzantine firing squads or (randomised) sublinear-time solutions for either problem exist. 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 can also obtain efficient randomised solutions, and it is straightforward to adapt our framework to allow f < n/2 omission or f < n crash faults.