Fast consensus in networks of bounded degree
In a Distributed Consensus protocol all processors (of which t may be faulty) are given (binary) initial values; after exchanging messages all correct processors must agree on one of them. In this paper we focus on consensus in networks that are not completely interconnected, following the work of Dwork et al. [DPPU]. 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+logn loglogn) one bit parallel transmission steps, while tolerating the asymptotically optimal number of faulty processors and asymptotically minimal number of exceptions. This result considerably improves upon our previous protocol, in particular it replaces the running time of O (n logn loglogn) with an asymptotically optimal one. As in [DPPU], we can 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 complete 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, with better constant factors than previously published results.
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