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Efficient matrix rank computation with application to the study of strongly regular graphs

Published: 29 July 2007 Publication History

Abstract

We present algorithms for computing the p-rank of integer matrices. They are designed to be particularly effective when p is a small prime, the rank is relatively low, and the matrix itself is large and dense and may exceed virtual memory space. Our motivation comes from the study of difference sets and partial difference sets in algebraic design theory. The p-rank of the adjacency matrix of an associated strongly regular graph is a key tool for distinguishing difference set constructions and thus answering various existence questions and conjectures. For the p-rank computation, we review several memory efficient methods, and present refinements suitable to the small prime, small rank case. We give a new heuristic approach that is notably effective in practice as applied to the strongly regular graph adjacency matrices. It involves projection to a matrix of order slightly above the rank. The projection is extremely sparse, is chosen according to one of several heuristics, and is combined with a small dense certifying component. Our algorithms and heuristics are implemented in the LinBox library. We also briefly discuss some of the software design issues and we present results of experiments for the Paley and Dickson sequences of strongly regular graphs.

References

[1]
M. Brookes. The matrix reference manual. http://www.ee.ic.ac.uk/hp/staff/www/matrix/property.html#schurcomp, 2005. {Online; accessed 22-January-2007}.
[2]
A. E. Brouwer and C. A. Van Eijl. On the p-rank of the adjacency matrices of strongly regular graphs. J. Algebraic Comb., 1(4):329--346, 1992.
[3]
L. Chen, W. Eberly, E. Kaltofen, W. Turner, B. D. Saunders, and G. Villard. Efficient matrix preconditioners for black box linear algebra. LAA 343--344, 2002, pages 119--146, 2002.
[4]
C. Ding and J. Yuan. A family of skew Hadamard difference set. J. Comb. Theory, Ser. A, 113:1526--1535, 2006.
[5]
J.-G. Dumas, T. Gautier, M. Giesbrecht, P. Giorgi, B. Hovinen, E. Kaltofen, B. D. Saunders, W. Turner, and G. Villard. Linbox: A generic library for exact linear algebra. In ICMS'02, pages 40--50, 2002.
[6]
J.-G. Dumas, T. Gautier, and C. Pernet. Finite field linear algebra subroutines. In Proc. of ISSAC'02, pages 63--74. ACM Press, 2002.
[7]
J.-G. Dumas, P. Giorgi, and C. Pernet. FFPACK: Finite field linear algebra package. In Proc. of ISSAC'05, pages 119--126, 2004.
[8]
J.-G. Dumas, B. D. Saunders, and G. Villard. Smith form via the valence: Experience with matrices from homology. In Proc. of ISSAC'00, pages 95--105. ACM Press, 2000.
[9]
J.-G. Dumas and G. Villard. Computing the rank of large sparse matrices over finite fields. In Proc. CASC'2002, The Fifth International Workshop on Computer Algebra in Scientific Computing, pages 22--27. Springer-Verlag, 2002.
[10]
J.-C. Faugère. Parallelization of Gröbner basis. In H. Hong, editor, PASCO'94, volume 5 of Lecture notes series in computing, pages 109--133, 1994.
[11]
Thierry Gautier, Jean-Louis Roch, and Gilles Villard. Givaro, a C++ for algebraic computations. http://www-lmc.imag.fr/Logiciels/givaro.
[12]
M. Giesbrecht, A. Lobo, and B.D. Saunders. Certifying inconsistency of sparse linear systems. In Proc. of ISSAC'98, pages 113--119. ACM Press, 1998.
[13]
E. Kaltofen. An output-sensitive variant of the baby steps/giant steps determinant algorithm. In Proc. of ISSAC'05, pages 138--144, 2002.
[14]
E. Kaltofen and B.D. Saunders. On Wiedemann's method of solving sparse linear systems. In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, volume 539 of LNCS, pages 29--38, 1991.
[15]
B.A. LaMacchia and A.M. Odlyzko. Solving large sparse linear systems over finite fields. Lecture Notes in Computer Science, 537:109--133, 1991.
[16]
S.L. Ma. A survey of partial difference sets. Designs, Codes and Cryptography, 4:221--261, 1994.
[17]
The LinBox Team. Linbox, a C++ library for exact linear algebra. http://www.linalg.org/.
[18]
W. Turner. Preconditioners for singular black box matrices. In Proc. of ISSAC'05, pages 332--339, New York, NY, USA, 2005. ACM Press.
[19]
W. Turner. A block wiedemann rank algorithm. In Proc. of ISSAC'06, pages 332--339, New York, NY, USA, 2006. ACM Press.
[20]
G. Weng, W. Qiu, Z. Wang, and Q. Xiang. Pseudo-paley graphs and skew Hadamard difference sets from commutative semifields, 2006. Preprint.
[21]
D. Wiedemann. Solving sparse linear equations over finite fields. IEEE Trans. Inform. Theory, 32:54--62, 1986.
[22]
Q. Xiang. Recent progress in algebraic design theory. Finite Fields and Their Applications, 11:622--653, 2005.

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  • (2013)CryptographyHandbook of Finite Fields10.1201/b15006-22(777-860)Online publication date: 17-Jun-2013
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cover image ACM Conferences
ISSAC '07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation
July 2007
406 pages
ISBN:9781595937438
DOI:10.1145/1277548
  • General Chair:
  • Dongming Wang
Permission to make digital or hard copies of all 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. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 29 July 2007

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  1. matrix p-rank
  2. out of core methods

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ISSAC07
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ISSAC07: International Symposium on Symbolic and Algebraic Computation
July 29 - August 1, 2007
Ontario, Waterloo, Canada

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Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

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  • (2015)3-ranks for strongly regular graphsProceedings of the 2015 International Workshop on Parallel Symbolic Computation10.1145/2790282.2790295(101-108)Online publication date: 10-Jul-2015
  • (2014)Linear independence oracles and applications to rectangular and low rank linear systemsProceedings of the 39th International Symposium on Symbolic and Algebraic Computation10.1145/2608628.2608673(381-388)Online publication date: 23-Jul-2014
  • (2013)CryptographyHandbook of Finite Fields10.1201/b15006-22(777-860)Online publication date: 17-Jun-2013
  • (2012) Commutative Semifields of Order 3 5 Communications in Algebra10.1080/00927872.2010.54427340:3(988-996)Online publication date: Mar-2012
  • (2012)Finite semifields with 74 elementsInternational Journal of Computer Mathematics10.1080/00207160.2012.68811389:13-14(1865-1878)Online publication date: 1-Sep-2012
  • (2011)New advances in the computational exploration of semifieldsInternational Journal of Computer Mathematics10.1080/00207160.2010.54851888:9(1990-2000)Online publication date: 1-Jun-2011
  • (2011)Simultaneous modular reduction and Kronecker substitution for small finite fieldsJournal of Symbolic Computation10.1016/j.jsc.2010.08.01546:7(823-840)Online publication date: 1-Jul-2011
  • (2009)Large matrix, small rankProceedings of the 2009 international symposium on Symbolic and algebraic computation10.1145/1576702.1576746(317-324)Online publication date: 28-Jul-2009

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