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.

我们使用排队网络来提出一种解决拉普拉斯系统的新方法。这标志着与现有技术的显着背离，这些技术主要基于图论构造和采样。我们的分布式求解器适用于一大类重要的拉普拉斯系统，我们称之为“单汇”拉普拉斯系统。具体地，我们的解算器可以产生以下形式的系统的解决方案\（L \ varvec {X} = \ varvec {B} \） ，其中的坐标中的正好一个\（\ varvec {B} \）是负的。我们的求解器是一个分布式算法，它需要\({\widetilde{O}}(t_{\text{ hit }}\hat{d}_{\max })\)时间（其中\({\widetilde{O} }\)隐藏\({\text {poly}}\log n\)因子）以产生近似解，其中\(t_{\text{ hit }}\)是图上随机游走的最坏情况击中时间，对于一大组重要图是\(\Theta (n)\)，而\(\ hat{d}_{\max }\)是图的最大度数。单汇拉普拉斯算子类包括重要的电压计算问题，并允许我们计算分布式设置中节点之间的有效电阻。因此，我们的拉普拉斯求解器可用于调整 Kelner 和 Mądry (2009) 的方法，以提供第一个分布式算法来有效地计算近似随机生成树。