Accelerated Distributed Laplacian Solvers via Shortcuts
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In this work we refine the analysis of the distributed Laplacian solver recently established by Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS '21), via the Ghaffari-Haeupler framework (SODA '16) of low-congestion shortcuts. Specifically, if ϵ > 0 represents the error of the solver, we derive two main results. First, for any n-node graph G with hop-diameter D and treewidth bounded by k, we show the existence of a Laplacian solver with round complexity O(n^o(1)kD log(1/ϵ)) in the CONGEST model. For graphs with bounded treewidth this circumvents the notorious Ω(√(n)) lower bound for "global" problems in general graphs. Moreover, following a recent line of work in distributed algorithms, we consider a hybrid communication model which enhances CONGEST with very limited global power in the form of the recently introduced node-capacitated clique. In this model, we show the existence of a Laplacian solver with round complexity O(n^o(1)log(1/ϵ)). The unifying thread of these results is an application of accelerated distributed algorithms for a congested variant of the standard part-wise aggregation problem that we introduce. This primitive constitutes the primary building block for simulating "local" operations on low-congestion minors, and we believe that this framework could be more generally applicable.
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