Tradeoffs between parallelism and fill in nested dissection

In this paper we demonstrate that parallelism and fill can be traded off in orders for Gaussian elimination. While the well-known nested dissection algorithm produces very parallel elimination orders, we show that by reducing the parallelism it is possible to reduce the fill that the orders generate. In particular, we present a new “less parallel nested dissection” algorithm (LPND). We prove that, unlike standard nested dissection, when applied to a chordal graph LPND finds a zero-fill elimination order. Our implementation of LPND generates less fill than state-of-the-art implementations of the nested dissection (METIS), minimum-degree @MD), and hybrid (BEND) algorithms on a large body of test matrices, at the cost of a small reduction in the paralellism in the orders that it produces. We have also implemented a nested dissection algorithm that is different from METIS and that uses the same separator algorithm used by our implementation of LPND. This algorithm, like LPND, generates less fill than METIS, and on large graphs generates significantly less fill than AMD. The latter comparison is notable, because although it is known that, for certain classes of graphs, minimum-degree produces asymptotically more fill than nested dissection, minimumdegree is believed to produce low-fill orderings in practice. Our experiments contradict this belief. ‘Universidade Federal do Rio de Janeiro, (claudson@acd.ufrj.br). Claudson was supported in part by NSF Grant CCR-9505472. *School of Computer Science, Carnegie Mellon University, and Akamai Technologies, Inc. Bruce Maggs is supported in part by the Air Force Materiel Command (AFMC) and DARPA under Contracts F19628-93-CO193 and F19628-96-G0061,by DARPA Contract NOOO14-95-1-1246, and by an NSF National Young Investigator Award, No. CCR-94-57766, with matching funds providedby NEC Research Institute and Sun Microsystems. 3Department of Computer Science, Carnegie Mellon University, (gImiIler@cs.cmu.edu). Gary Miller issupportedin part by NSF Grants CCR-9505472 and CCR-9706572. Permission to make digital or hard copies ofall or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists. requires prior spccitic permission and/or a fee. SPAA ‘99 Saint Malo, France Copyright ACM 1999 l-581 13-124-O/99/06.,.$5.00