Matrix sparsification for rank and determinant computations via nested dissection

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Abstract

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.

Original languageEnglish
Title of host publicationProceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
Pages137-145
Number of pages9
DOIs
StatePublished - 2008
Event49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States
Duration: 25 Oct 200828 Oct 2008

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
Country/TerritoryUnited States
CityPhiladelphia, PA
Period25/10/0828/10/08

ASJC Scopus subject areas

  • General Computer Science

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