On Computing Min-Degree Elimination Orderings

Abstract: We study faster algorithms for producing the minimum degree ordering used to speed up Gaussian elimination. This ordering is based on viewing the non-zero elements of a symmetric positive definite matrix as edges of an undirected graph, and aims at reducing the additional non-zeros (fill) in the matrix by repeatedly removing the vertex of minimum degree. It is one of the most widely used primitives for pre-processing sparse matrices in scientific computing. Our result is in part motivated by the observation that sub-quadratic time algorithms for finding min-degree orderings are unlikely, assuming the strong exponential time hypothesis (SETH). This provides justification for the lack of provably efficient algorithms for generating such orderings, and leads us to study speedups via degree-restricted algorithms as well as approximations. Our two main results are: (1) an algorithm that produces a min-degree ordering whose maximum degree is bounded by $\Delta$ in $O(m \Delta \log^3{n})$ time, and (2) an algorithm that finds an $(1 + \epsilon)$-approximate marginal min-degree ordering in $O(m \log^{5}n \epsilon^{-2})$ time. Both of our algorithms rely on a host of randomization tools related to the $\ell_0$-estimator by [Cohen `97]. A key technical issue for the final nearly-linear time algorithm are the dependencies of the vertex removed on the randomness in the data structures. To address this, we provide a method for generating a pseudo-deterministic access sequence, which then allows the incorporation of data structures that only work under the oblivious adversary model.