Sparsified Cholesky Solvers for SDD linear systems

We show that Laplacian and symmetric diagonally dominant (SDD) matrices can be well approximated by linear-sized sparse Cholesky factorizations. Specifically, n × n matrices of these types have constant-factor approximations of the form LL , where L is a lowertriangular matrix with O(n) non-zero entries. This factorization allows us to solve linear systems in such matrices in O(n) work and O(log n log log n) depth. We also present nearly linear time algorithms that construct solvers that are almost this efficient. In doing so, we give the first nearly-linear work routine for constructing spectral vertex sparsifiers—that is, spectral approximations of Schur complements of Laplacian matrices.