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

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 $\tilde{O}(t_{hit}{d}_{\max })$ time (where $\tilde{O}$ 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 Mądry (2009) to give the first distributed algorithm to compute approximate random spanning trees efficiently.

References in corpus (3)

• Current Flow Group Closeness Centrality for Complex Networks
• A Fast Distributed Solver for Symmetric Diagonally Dominant Linear Equations
• A New Approach to Laplacian Solvers and Flow Problems