Matrix Sparsification for Rank and Determinant Computations via Nested Dissection

The nested dissection method developed by Lipton, Rose, and Tarjan is a seminal method for quickly performing Gaussian elimination of symmetric real positive definite matrices whose support structure satisfies good separation properties (e.g. planar). One can use the resulting LU factorization to deduce various parameters of the matrix. The main results of this paper show that we can remove the three restrictions of being "symmetric", being "real", and being "positive definite" and still be able to compute the rank and, when relevant, also the absolute determinant, while keeping the running time of nested dissection. Our results are based, in part, on an algorithm that, given an arbitrary square matrix A of order n having m non-zero entries, creates another square matrix B of order n + 2t = O(m) with the property that each row and each column of B contains at most three nonzero entries, and, furthermore, rank(B) = rank (A) + 2t and det(B) = det(A). The running time of this algorithm is only O(m), which is optimal.