A Queueing Network-Based Distributed Laplacian Solver

We use queueing networks to present a new approach to solving Laplacian systems. This marks a significant departure from the existing techniques, mostly based on graph-theoretic constructions and sampling. Our distributed solver works for a large and important class of Laplacian systems that we call “one-sink” Laplacian systems. Specifically, our solver can produce solutions for systems of the form $$L\varvec{x} = \varvec{b}$$ where exactly one of the coordinates of $$\varvec{b}$$ is negative. Our solver is a distributed algorithm that takes $${\widetilde{O}}(t_{\text{ hit }}\hat{d}_{\max })$$ time (where $${\widetilde{O}}$$ hides $${\text {poly}}\log n$$ factors) to produce an approximate solution where $$t_{\text{ hit }}$$ is the worst-case hitting time of the random walk on the graph, which is $$\Theta (n)$$ for a large set of important graphs, and $$\hat{d}_{\max }$$ is the maximum degree of the graph. The class of one-sink Laplacians includes the important voltage computation problem and allows us to compute the effective resistance between nodes in a distributed setting. As a result, our Laplacian solver can be used to adapt the approach by Kelner and Mądry (2009) to give the first distributed algorithm to compute approximate random spanning trees efficiently.