Nearly-Linear Work Parallel SDD Solvers, Low-Diameter Decomposition, and Low-Stretch Subgraphs

Abstract

We present the design and analysis of a nearly-linear work parallel algorithm for solving symmetric diagonally dominant (SDD) linear systems. On input an SDD n-by-n matrix A with m nonzero entries and a vector b, our algorithm computes a vector $\tilde{x}$ such that $\|\tilde{x} - A^{+}b\|_{A} \leq\varepsilon\cdot\|{A^{+}b}\|_{A}$ in $O(m\log^{O(1)}{n}\log {\frac{1}{\varepsilon}})$ work and $O(m^{1/3+\theta}\log\frac{1}{\varepsilon})$ depth for any θ>0, where A ⁺ denotes the Moore-Penrose pseudoinverse of A.

The algorithm relies on a parallel algorithm for generating low-stretch spanning trees or spanning subgraphs. To this end, we first develop a parallel decomposition algorithm that in O(mlog^O(1) n) work and polylogarithmic depth, partitions a graph with n nodes and m edges into components with polylogarithmic diameter such that only a small fraction of the original edges are between the components. This can be used to generate low-stretch spanning trees with average stretch O(n ^α) in O(mlog^O(1) n) work and O(n ^α) depth for any α>0. Alternatively, it can be used to generate spanning subgraphs with polylogarithmic average stretch in O(mlog^O(1) n) work and polylogarithmic depth. We apply this subgraph construction to derive a parallel linear solver.

By using this solver in known applications, our results imply improved parallel randomized algorithms for several problems, including single-source shortest paths, maximum flow, minimum-cost flow, and approximate maximum flow.