Fast consensus in networks of bounded degree

The Distributed Consensus problem involves n processors each of which holds an initial binary value. At most t of the processors may be faulty and ignore any protocol (even behaving maliciously), yet it is required that the non-faulty processors eventually agree on a value that was initially held by one of them. In this paper we focus on consensus in networks whose degree is bounded, following the work of Dwork, Peleg, Pippenger and Upfal [8]. In such a context, complete consensus among all the correct processors is not possible and some exceptions must be allowed. We first show how to achieve consensus in the butterfly network using O(t + log n loglog n) one-bit parallel transmission steps, while tolerating the asymptotically optimal number of faulty processors (O(n/log n)) and having the asymptotically minimal number of exceptions (O(t log t)). This result considerably improves on the running time of existing butterfly consensus protocols [2, 8]. In particular, it replaces the running time of O(n log n loglog n) of [2] with an asymptotically optimal one. As in [8], we can then decrease the number of exceptions to O(t) by using additional links, while maintaining the same running time. The protocol is derived from a consensus protocol for completely connected networks that is interesting in its own right: it achieves Distributed Consensus with optimal number of processors, asymptotically optimal total bit transfer and nearly optimal number of rounds.