  We describe a simple algorithm for spectral graph sparsification, based on
iterative computations of weighted spanners and uniform sampling. Leveraging
the algorithms of Baswana and Sen for computing spanners, we obtain the first
distributed spectral sparsification algorithm. We also obtain a parallel
algorithm with improved work and time guarantees. Combining this algorithm with
the parallel framework of Peng and Spielman for solving symmetric diagonally
dominant linear systems, we get a parallel solver which is much closer to being
practical and significantly more efficient in terms of the total work.
