A Distributed Laplacian Solver and its Applications to Electrical Flow and Random Spanning Tree Computation

TL;DR

This paper introduces a novel distributed algorithm for solving a specific class of Laplacian systems using queueing networks, enabling efficient electrical flow computations and random spanning tree generation in large graphs.

Contribution

It presents the first distributed solver for one-sink Laplacian systems, leveraging queueing networks instead of traditional graph-theoretic methods.

Findings

Solver runs in ( t_{hit} d_{max}) time for approximate solutions.

Applicable to voltage computation and effective resistance in distributed settings.

Enables efficient distributed approximation of random spanning trees.

Abstract

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 $Lx = b$ where exactly one of the coordinates of $b$ is negative. Our solver is a distributed algorithm that takes $\widetilde{O}(t_{hit} d_{\max})$ time (where $\widetilde{O}$ hides $\text{poly}\log n$ factors) to produce an approximate solution where $t_{hit}$ 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 $d_{\max}$ is the generalized 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 M\k{a}dry (2009) to give the first distributed algorithm to compute approximate random spanning trees efficiently.