Simple Parallel and Distributed Algorithms for Spectral Graph Sparsification

We describe simple algorithms for spectral graph sparsification, based on iterative computations of weighted spanners and sampling. Leveraging the algorithms of Baswana and Sen for computing spanners, we obtain the first distributed spectral sparsification algorithm in the CONGEST model. We also obtain a parallel algorithm with improved work and time guarantees, as well as other natural distributed implementations. Combining this algorithm with the parallel framework of Peng and Spielman for solving symmetric diagonally dominant linear systems, we get a parallel solver that is significantly more efficient in terms of the total work.