Efficient counting with optimal resilience

Abstract

Consider a complete communication network of n nodes, where the nodes receive a common clock pulse. We study the synchronous c-counting problem: given any starting state and up to f faulty nodes with arbitrary behavior, the task is to eventually have all correct nodes labeling the pulses with increasing values modulo c in agreement. Thus, we are considering algorithms that are self-stabilizing despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have optimal resilience, (3) have a linear stabilization time in f (asymptotically optimal), (4) use a small number of states, and, consequently, (5) communicate a small number of bits per round. Prior algorithms either resort to randomization, use a large number of states and need high communication bandwidth, or have suboptimal resilience. In particular, we achieve an exponential improvement in both state complexity and message size for deterministic algorithms. Moreover, we present twocomplementary approaches for reducing the number of bits communicated during and after stabilization.