Tradeoos between Parallelism and Fill in Nested Dissection

In this paper we demonstrate that tradeoos can be made between parallelism and ll in nested dissection algorithms for Gaussian elimination, both in theory and in practice. We present a new \less parallel nested dissection" algorithm (LPND), and prove that, unlike the standard nested dissection algorithm, when applied to a chordal graph LPND nds a zero-ll elimination order. We have also implemented the LPND algorithm. On a variety of benchmarks it generates less ll than state-of-the-art implementations of the nested dissection (METIS), minimum-degree (AMD), and hybrid (BEND) algorithms on a large body of test matrices. We have also implemented another nested dissection algorithm that is diierent from METIS and that uses the same separator algorithm used by our implementation of LPND. This algorithm, as well as LPND, generates less ll than METIS, and on large graphs signiicantly outperforms AMD. The latter comparison is notable, because although it is known that, for certain classes of graphs, minimum-degree produces asymptotically more ll than nested dissection, minimum-degree is believed to produce low-ll orderings in practice. Our experiments contradict this belief.