Byzantine Agreement (BA) is a key component in many distributed systems. While Dolev and Reischuk have proven a long time ago that quadratic communication complexity is necessary for worst-case runs, the question of what can be done in practically common runs with fewer failures remained open. In this paper we present the first Byzantine Broadcast algorithm with O(n(f+1)) communication complexity in a model with resilience of n = 2t+1, where 0 ≤ f ≤ t is the actual number of process failures in a run. And for BA with strong unanimity, we present the first optimal-resilience algorithm that has linear communication complexity in the failure-free case and a quadratic cost otherwise.
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Ittai Abraham (2018) studied this question.
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