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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 didnot report experimental results. We seek to both evaluate the performance ofthis approach and to explore improvements to it in practice. We compare theperformance of this method to other Laplacian solvers on a variety of realworld graphs. We consider different ways to improve the performance of thismethod by exploring different ways of choosing the set of cycles and thesequence of updates, with the goal of providing more flexibility and potentialparallelism. We propose a parallel model of the Kelner et al. method, forevaluating potential parallelism in terms of the span of edges updated at eachiteration. We provide experimental results comparing the potential parallelismof the fundamental cycle basis and our extended cycle set. Our preliminaryexperiments show that choosing a non-fundamental set of cycles can savesignificant work compared to a fundamental cycle basis.

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