Location-Aided Fast Distributed Consensus

Existing works on distributed averaging explore linear iterations based on reversible Markov chains. The convergence of such algorithms is bounded to be slow due to the diffusive behavior of the reversible chains. It has been observed that by overcoming the diffusive behavior, certain nonreversible chains lifted from reversible ones mix substantially faster than the original chains [1], [2]. In this paper, we investigate the idea of fast distributed consensus via lifting Markov chains, and propose a class of Location-Aided Distributed Averaging (LADA) algorithms for wireless networks, where nodes’ location information is used to construct nonreversible chains that facilitates distributed computing and cooperative processing. We first show that it is possible to achieve an ǫ-averaging time of O(r log(ǫ)) in a wireless network with transmission radius r with a centralized algorithm, which is close to a performance lower bound derived based on resistance, an invariant of Markov chains. We then present a distributed LADA algorithm, which utilizes only the direction information of neighbors to construct nonreversible chains. The constructed chain does not naturally possess a uniform stationary distribution, which is in turn compensated by a weight estimation procedure to yield the average estimate. It is shown that LADA achieves the same scaling law in averaging time as the centralized scheme in wireless networks for all r satisfying the connectivity requirement. Finally, we propose a cluster-based LADA (C-LADA) algorithm, which requires no central coordination for clustering, and provides the additional benefit of reduced message complexity compared with LADA.