Evaluating the Potential of a Dual Randomized Kaczmarz Solver for Laplacian Linear Systems

TL;DR

This paper evaluates a randomized cycle-based solver for Laplacian systems, compares its performance with other methods, and explores improvements including cycle selection strategies and parallelism potential.

Contribution

It provides experimental evaluation of Kelner et al.'s method, introduces a parallel model, and demonstrates that alternative cycle sets can improve efficiency.

Findings

Choosing non-fundamental cycle sets reduces computational work.

Parallel model reveals potential for increased efficiency.

Experimental results show improved performance with cycle set modifications.

Abstract

A new method for solving Laplacian linear systems proposed by Kelner et al. involves the random sampling and update of fundamental cycles in a graph. Kelner et al. proved asymptotic bounds on the complexity of this method but did not report experimental results. We seek to both evaluate the performance of this approach and to explore improvements to it in practice. We compare the performance of this method to other Laplacian solvers on a variety of real world graphs. We consider different ways to improve the performance of this method by exploring different ways of choosing the set of cycles and the sequence of updates, with the goal of providing more flexibility and potential parallelism. We propose a parallel model of the Kelner et al. method, for evaluating potential parallelism in terms of the span of edges updated at each iteration. We provide experimental results comparing the potential parallelism of the fundamental cycle basis and our extended cycle set. Our preliminary experiments show that choosing a non-fundamental set of cycles can save significant work compared to a fundamental cycle basis.