A separator theorem for graphs of bounded genus
Journal of Algorithms
Matrix analysis
Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
The analysis of a nested dissection algorithm
Numerische Mathematik
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Fast construction of irreducible polynomials over finite fields
Journal of Symbolic Computation
A Linear Time Algorithm for Embedding Graphs in an Arbitrary Surface
SIAM Journal on Discrete Mathematics
Determinant algorithms for random planar structures
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Shallow excluded minors and improved graph decompositions
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Some optimal inapproximability results
Journal of the ACM (JACM)
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Spectral partitioning works: planar graphs and finite element meshes
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Solving sparse rational linear systems
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Maximum matching in graphs with an excluded minor
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Matrix Sparsification for Rank and Determinant Computations via Nested Dissection
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
A linear-time algorithm to find a separator in a graph excluding a minor
ACM Transactions on Algorithms (TALG)
A Separator Theorem in Minor-Closed Classes
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Solving Linear Systems through Nested Dissection
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
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The generalized nested dissection method, developed by Lipton et al. [1979], is a seminal method for solving a linear system Ax=b where A is a symmetric positive definite matrix. The method runs extremely fast whenever A is a well-separable matrix (such as matrices whose underlying support is planar or avoids a fixed minor). In this work, we extend the nested dissection method to apply to any nonsingular well-separable matrix over any field. The running times we obtain essentially match those of the nested dissection method. An important tool is a novel method for matrix sparsification that preserves determinants and minors, and that guarantees that constant powers of the sparsified matrix remain sparse.