Exploiting Spontaneous Transmissions for Broadcasting and Leader Election in Radio Networks

We study two fundamental communication primitives: broadcasting and leader election in the classical model of multi-hop radio networks with unknown topology and without collision detection mechanisms. It has been known for almost 20 years that in undirected networks with n nodes and diameterD, randomized broadcasting requiresΩ(D log n D +log n) rounds in expectation, assuming that uninformed nodes are not allowed to communicate (until they are informed). Only very recently, Haeupler and Wajc (PODC’2016) showed that this bound can be slightly improved for the model with spontaneous transmissions, providing an O(D log n log logn logD + log O(1) n)-time broadcasting algorithm. In this paper, we give a new and faster algorithm that completes broadcasting in O(D logn logD + log O(1) n) time, with high probability. This yields the first optimal O(D)-time broadcasting algorithm whenever D is polynomial in n. Furthermore, our approach can be applied to design a new leader election algorithm that matches the performance of our broadcasting algorithm. Previously, all fast randomized leader election algorithms have been using broadcasting as their subroutine and their complexity have been asymptotically strictly bigger than the complexity of broadcasting. In particular, the fastest previously known randomized leader election algorithm of Ghaffari and Haeupler (SODA’2013) requiresO(D log n D min{log logn, log n D }+ log n)-time with high probability. Our new algorithm requiresO(D logn logD+log O(1) n) time with high probability, and it achieves the optimalO(D) time wheneverD is polynomial in n. Research partially supported by the Centre for Discrete Mathematics and its Applications (DIMAP). Contact information: {A.Czumaj, P.W.Davies}@warwick.ac.uk. Phone: +44 24 7657 3796.