Article

Dynamic Normal Forms and Dynamic Characteristic Polynomial

Authors:

Gudmund Skovbjerg Frandsen,

Piotr SankowskiAuthors Info & Claims

ICALP '08: Proceedings of the 35th international colloquium on Automata, Languages and Programming - Volume Part I

Pages 434 - 446

https://doi.org/10.1007/978-3-540-70575-8_36

Published: 07 July 2008 Publication History

Abstract

We present the first fully dynamic algorithm for computing thecharacteristic polynomial of a matrix. In the generic symmetriccase our algorithm supports rank-one updates inO(n²logn) randomized timeand queries in constant time, whereas in the general case thealgorithm works in O(n²klogn) randomized time, where kis the number ofinvariant factors of the matrix. The algorithm is based on thefirst dynamic algorithm for computing normal forms of a matrix suchas the Frobenius normal form or the tridiagonal symmetric form. Thealgorithm can be extended to solve the matrix eigenproblem withrelative error 2^-bin additionalO(nlog²nlogb)time. Furthermore, it can be used to dynamically maintain thesingular value decomposition (SVD) of a generic matrix. Togetherwith the algorithm the hardness of the problem is studied. For thesymmetric case we present an Ω(n²) lower bound for rank-one updates and anΩ(n) lower bound for element updates.

References

[1]

Giesbrecht, M.: Nearly optimal algorithms for canonical matrix forms. SIAM Journal on Computing 24(5), 948-969 (1995)

Digital Library

[2]

Eberly, W.: Asymptotically efficient algorithms for the Frobenius form. Paper 723- 26, Department of Computer Science, University of Calgary (2003)

[3]

Villard, G.: Computing the Frobenius normal form of a sparse matrix. In: The Third International Workshop on Computer Algebra in Scientific Computing, pp. 395-407. Springer, Heidelberg (2000)

[4]

Storjohann, A.: Deterministic computation of the frobenius form. In: FOCS, pp. 368-377 (2001)

Digital Library

[5]

Ben-Amram, A., Galil, Z.: On pointers versus addresses. J. Assoc. Comput. Mach. 39, 617-648 (1992)

Digital Library

[6]

Pan, V., Chen, Z.: The complexity of the matrix eigenproblem. In: STOC 1999: Proceedings of the thirty-first annual ACM symposium on Theory of computing, pp. 507-516. ACM Press, New York (1999)

Digital Library

[7]

Frandsen, G., Hansen, J., Miltersen, P.: Lower bounds for dynamic algebraic problems. Inform. and Comput. 171(2), 333-349 (2001)

Digital Library

[8]

Frandsen, P., Frandsen, G.: Dynamic matrix rank. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 395-406. Springer, Berlin (2006)

[9]

Sankowski, P.: Dynamic Transitive Closure via Dynamic Matrix Inverse. In: FOCS, pp. 509-517 (2004)

Digital Library

[10]

Ibarra, O., Moran, S., Rosier, L.: A note on the parallel complexity of computing the rank of order n matrices. Inform. Process. Lett. 11(4-5), 162 (1980)

[11]

Mulmuley, K.: A fast parallel algorithm to compute the rank of a matrix over an arbitrary field. In: STOC, pp. 338-339. ACM Press, New York (1986)

Digital Library

[12]

Chung, F.R.K.: Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, vol. 92. American Mathematical Society (1997)

[13]

Babai, L., Grigoryev, D., Mount, D.: Isomorphism of graphs with bounded eigenvalue multiplicity. In: STOC 1982: Proceedings of the fourteenth annual ACM symposium on Theory of computing, New York, NY, USA, pp. 310-324 (1982)

Digital Library

[14]

Fiedler, M.: Algebraic connectivity of graphs. Czechoslovak Mathematical Journal 23(98), 289-305 (1973)

[15]

Spielman, D., Teng, S.H.: Spectral partitioning works: Planar graphs and finite element meshes. In: FOCS, pp. 96-105 (1996)

Digital Library

[16]

Weiss, Y.: Segmentation using eigenvectors: A unifying view. In: ICCV (2), pp. 975-982 (1999)

Digital Library

[17]

Ng, A., Jordan, M., Weiss, Y.: On spectral clustering: Analysis and an algorithm. In: Proceedings of Advances in Neural Information Processing Systems 14 (2001)

[18]

Meyer, C.D., J.S.: Updating finite markov chains by using techniques of group matrix inversion. J. Statist. Comput. Simulat. 11, 163-181 (1980)

[19]

Funderlic, R.E., Plemmons, R.J.: Updating lu factorizations for computing stationary distributions. SIAM J. Algebraic Discrete Methods 7(1), 30-42 (1986)

Digital Library

[20]

Seneta, E.: Sensivity analysis, ergodicity coefficients, and rank-one updates for finite markov chains. Numerical Solutions of Markov Chains, 121-129 (1991)

[21]

Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1(1), 36-61 (1991), http://www.ams.org/mathscinet-getitem?mr=92h:60103

[22]

Golub, G.H., Van Loan, C.F.: Matrix computations, 3rd edn. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1996)

Digital Library

[23]

Chandrasekaran, S., Manjunath, B.S., Wang, Y.F., Winkeler, J., Zhang, H.: An eigenspace update algorithm for image analysis. Graph. Models Image Process 59(5), 321-332 (1997)

Digital Library

[24]

Kanth, K., Agrawal, D., Singh, A.: Dimensionality reduction for similarity searching in dynamic databases. In: Proceedings of the 1998 ACM SIGMOD international conference on Management of data, NY, USA, pp. 166-176 (1998)

Digital Library

[25]

Brand, M.: Fast online svd revisions for lightweight recommender systems. In: Barbará, D., Kamath, C. (eds.) SDM. SIAM, Philadelphia (2003)

[26]

Gu, M., Eisenstat, S.C.: A stable and fast algorithm for updating singular value decomposition. Technical Report YALE/DCS/TR-966, Yale University, New Haven, CT (1993)

[27]

Gu, M., Eisenstat, S.: Downdating the singular value decomposition. SIAM Journal on Matrix Analysis and Applications 16(3), 793-810 (1995)

Digital Library

[28]

Frandsen, G.S., Sankowski, P.: Dynamic normal forms and dynamic characteristic polynomial. Research Series RS-08-2, BRICS, Department of Computer Science, University of Aarhus (2008), http://www.brics.dk/RS/08/2/index.html

[29]

Bürgisser, P., Clausen, M., Shokrollahi, M.: Algebraic complexity theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 315. Springer, Berlin (1997)

[30]

Bini, D., Pan, V.: Polynomial and Matrix Computations. Birkhäuser (1994)

Digital Library

[31]

Eberly, W., Kaltofen, E.: On randomized lanczos algorithms. In: ISSAC 1997: Proceedings of the 1997 international symposium on Symbolic and algebraic computation, pp. 176-183. ACM Press, New York (1997)

Digital Library

Cited By

Sankowski PWęgrzycki K(2019)Improved Distance Queries and Cycle Counting by Frobenius Normal FormTheory of Computing Systems10.1007/s00224-018-9894-x63:5(1049-1067)Online publication date: 1-Jul-2019
https://dl.acm.org/doi/10.1007/s00224-018-9894-x
Kavitha T(2018)Dynamic Matrix Rank with Partial LookaheadTheory of Computing Systems10.1007/s00224-014-9531-255:1(229-249)Online publication date: 26-Dec-2018
https://dl.acm.org/doi/10.1007/s00224-014-9531-2
Frandsen GFrandsen P(2009)Dynamic matrix rankTheoretical Computer Science10.1016/j.tcs.2009.06.012410:41(4085-4093)Online publication date: 1-Sep-2009
https://dl.acm.org/doi/10.1016/j.tcs.2009.06.012
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Published In

cover image Guide Proceedings

ICALP '08: Proceedings of the 35th international colloquium on Automata, Languages and Programming - Volume Part I

July 2008

892 pages

ISBN:3540705740

Editors:
Luca Aceto
School of Computer Science, Reykjavik University, Reykjavík, Iceland 103
,
Ivan Damgård
Department of Computer Science, IT-Parken, University of Aarhus, ÅrhusN, Denmark 8200
,
Leslie Ann Goldberg
Department of Computer Science, University of Liverpool, Liverpool, UK L69 3BX
,
Magnús M. Halldórsson
School of Computer Science, Reykjavik University, Reykjavik, Iceland 103
,
Anna Ingólfsdóttir
Department of Computer Science, Reykjavik University, Reykjavík, Iceland 103
,
Igor Walukiewicz
Université de Bordeaux-1, LaBRI, Talence cedex, France 33405

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 07 July 2008

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Cited By

Sankowski PWęgrzycki K(2019)Improved Distance Queries and Cycle Counting by Frobenius Normal FormTheory of Computing Systems10.1007/s00224-018-9894-x63:5(1049-1067)Online publication date: 1-Jul-2019
https://dl.acm.org/doi/10.1007/s00224-018-9894-x
Kavitha T(2018)Dynamic Matrix Rank with Partial LookaheadTheory of Computing Systems10.1007/s00224-014-9531-255:1(229-249)Online publication date: 26-Dec-2018
https://dl.acm.org/doi/10.1007/s00224-014-9531-2
Frandsen GFrandsen P(2009)Dynamic matrix rankTheoretical Computer Science10.1016/j.tcs.2009.06.012410:41(4085-4093)Online publication date: 1-Sep-2009
https://dl.acm.org/doi/10.1016/j.tcs.2009.06.012
Sankowski P(2008)Algebraic Graph AlgorithmsProceedings of the 33rd international symposium on Mathematical Foundations of Computer Science10.1007/978-3-540-85238-4_5(68-82)Online publication date: 25-Aug-2008
https://dl.acm.org/doi/10.1007/978-3-540-85238-4_5

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