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Existing works on distributed consensus explore linear iterations based on reversible Markov chains, which contribute to the slow convergence of the algorithms. It has been observed that by overcoming the diffusive behavior of reversible chains, certain nonreversible chains lifted from reversible ones mix substantially faster than the original chains. In this paper, the idea of Markov chain lifting is studied to accelerate the convergence of distributed consensus, and two general pseudoalgorithms are presented. These pseudoalgorithms are then instantiated through a class of location-aided distributed averaging (LADA) algorithms for wireless networks, where nodes' coarse location information is used to construct nonreversible chains that facilitate distributed computing and cooperative processing. Our first LADA algorithm is designed for grid networks; for a k × k grid network, it achieves an ε-averaging time of O(k log(ε-1)). Based on this algorithm, in a wireless network with transmission range r, an ε-averaging time of O(r-1 log(ε-1)) can be attained through a centralized algorithm. Subsequently, a distributed LADA algorithm is presented, achieving the same scaling law in averaging time as the centralized scheme in wireless networks for all r satisfying the connectivity requirement; the constructed chain also attains the optimal scaling law in terms of an important mixing metric, the fill time, in its class. Finally, a cluster-based LADA algorithm is proposed, which, requiring no central coordination, provides the additional benefit of reduced message complexity compared with the distributed LADA algorithm.
Date of Publication: Dec. 2010