We present the first parallel algorithm for solving systems of linear
equations in symmetric, diagonally dominant (SDD) matrices that runs in
polylogarithmic time and nearly-linear work. The heart of our algorithm is a
construction of a sparse approximate inverse chain for the input matrix: a
sequence of sparse matrices whose product approximates its inverse. Whereas
other fast algorithms for solving systems of equations in SDD matrices exploit
low-stretch spanning trees, our algorithm only requires spectral graph
sparsifiers