TY - GEN

T1 - Matrix sparsification for rank and determinant computations via nested dissection

AU - Yuster, Raphael

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=57949095270&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2008.14

DO - 10.1109/FOCS.2008.14

M3 - Conference contribution

AN - SCOPUS:57949095270

SN - 9780769534367

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 137

EP - 145

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

T2 - 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

Y2 - 25 October 2008 through 28 October 2008

ER -