Advertisement

computational complexity

, Volume 1, Issue 3, pp 235–256 | Cite as

Existence and efficient construction of fast Fourier transforms on supersolvable groups

  • Ulrich Baum
Article

Abstract

The linear complexityLK(A) of a matrixA over a fieldK is defined as the minimal number of additions, subtractions and scalar multiplications sufficient to evaluateA at a generic input vector. IfG is a finite group andK a field containing a primitive exp(G)-th root of unity,LK(G):= min{LK(A)|A a Fourier transform forKG} is called theK-linear complexity ofG. We show that every supersolvable groupG has amonomial Fourier Transform adapted to a chief series ofG. The proof is constructive and gives rise to an efficient algorithm with running timeO(|G|2log|G|). Moreover, we prove that these Fourier transforms are efficient to evaluate:LK(G)≤8.5|G|log|G| for any supersolvable groupG andLK(G)≤1.5|G|log|G| for any 2-groupG.

Key words

fast Fourier transforms linear complexity supersolvable groups monomial representations 

Subject classifications

20C15 20C40 68Q40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M.D. Atkinson (ed.),Computational Group Theory, Academic Press, 1984.Google Scholar
  2. [2]
    L. Babai, L. Rónyai, Computing Irreducible Representations of Finite Groups,Proc. 30th IEEE Symp. Foundations of Comput. Science, 1989, 93–98.Google Scholar
  3. [3]
    U. Baum,Existenz und effiziente Konstruktion schneller Fouriertransformationen überauflösbarer Gruppen, Dissertation, Universität Bonn, 1991.Google Scholar
  4. [4]
    U. Baum, M. Clausen, Some Lower and Upper Complexity Bounds for Generalized Fourier Transforms and Their Inverses,SIAM J. Comput.,2 (1991), 451–459.Google Scholar
  5. [5]
    U. Baum, M. Clausen, B. Tietz, Improved Upper Complexity Bounds for the Discrete Fourier Transform,AAECC,2 (1991), 35–43.Google Scholar
  6. [6]
    T. Beth,Verfahren der schnellen Fourier-Transformation, Teubner, 1984.Google Scholar
  7. [7]
    T. Beth, On the Computational Complexity of the General Discrete Fourier Transform,Theor. Comp. Sci.,51 (1987), 331–339.Google Scholar
  8. [8]
    L.I. Bluestein, A Linear Filtering Approach to the Computation of the Discrete Fourier Transform,IEEE Trans.,AU 18 (1970), 451–455.Google Scholar
  9. [9]
    J. Bruck, Harmonic Analysis of Polynomial Threshold Functions,SIAM J. Disc. Math.,3 (1990), 168–177.Google Scholar
  10. [10]
    G. Butler, J. Cannon,Introduction to Computational Group Theory, Computers & Mathematics, Tutorial Minicourse Notes, MIT, Cambridge, MA, 1989.Google Scholar
  11. [11]
    J. Cannon, An Introduction to the Group Theory Language CAYLEY, in: [1] 145–183.Google Scholar
  12. [12]
    M. Clausen,Beiträge zum Entwurf schneller Spektraltransformationen, Habilitationsschrift, Universität Karlsruhe, 1988.Google Scholar
  13. [13]
    M. Clausen, Fast Fourier Transforms for Metabelian Groups,SIAM J. Comput.,18 (1989), 584–593.Google Scholar
  14. [14]
    M. Clausen, Fast Generalized Fourier Transforms,Theor. Comp. Sci.,67 (1989), 55–63.Google Scholar
  15. [15]
    J.W. Cooley, J.W., Tukey, An Algorithm for the Machine Calculation of Complex Fourier Series,Math. Comp.,19 (1965), 297–301.Google Scholar
  16. [16]
    C.W. Curtis, I. Reiner,Representation Theory of Finite Groups and Associative Algebras, Wiley & Sons, 1962.Google Scholar
  17. [17]
    P. Diaconis, A Generalization of Spectral Analysis with Application to Ranked Data,Ann. Statistics,17 (1989), 949–979.Google Scholar
  18. [18]
    P. Diaconis,Group Representations in Probability and Statistics, Institute of Mathematical Statistics Lecture Notes-Monograph Series,11, Hayward, CA, 1988.Google Scholar
  19. [19]
    G. Havas, M.F. Newman, Calculating Presentations for Certain Kinds of Quotient Groups,Proc. Assoc. Comput. Mach. SYMSAC'76, 1976, 2–8.Google Scholar
  20. [20]
    B. Huppert Endliche Gruppen I, Springer, 1967.Google Scholar
  21. [21]
    B. Huppert, N. Blackburn,Finite Groups II, Springer, 1982.Google Scholar
  22. [22]
    S.L. Hurst, D.M., Miller, J.C. Muzio,Spectral Techniques in Digital Logic, Academic Press, 1985.Google Scholar
  23. [23]
    G.J. Janusz, Primitive Idempotents in Group Algebras,Proc. Am. Math. Soc.,17 (1966), 520–523.Google Scholar
  24. [24]
    M.G. Karpovsky (ed.),Spectral Techniques and Fault Detection, Academic Press, 1985.Google Scholar
  25. [25]
    D.J. Klein, C.H. Carlisle, F.A. Matsen, Symmetry Adaptation to Sequences of Finite Groups,Advances in Quantum Chemistry,5 (1970), Academic Press, 219–260.Google Scholar
  26. [26]
    C.R. Leedham-Green, A Soluble Group Algorithm, in: [1], 85–101.Google Scholar
  27. [27]
    N. Linial, Y. Mansour, N. Nisan, Constant Depth Circuits, Fourier Transform, and Learnability,Proc. 30th IEEE Symp. Foundations of Comput. Science, 1989, 574–579.Google Scholar
  28. [28]
    J.D. Lipson,Elements of Algebra and Algebraic Computing, Benjamin-Cummings, 1981.Google Scholar
  29. [29]
    J. Morgenstern, Note on a Lower Bound of the Linear Complexity of the Fast Fourier Transform,J. Assoc. Comput. Mach.,20/2 (1973), 305–306.Google Scholar
  30. [30]
    K. Murota, K. Ikeda, Computational Use of Group Theory in Bifurcation Analysis of Symmetric Structures,SIAM J. Sci. Stat. Comput.,12 (1991), 273–297.Google Scholar
  31. [31]
    K. Niederdrenk,Die endliche Fourier- und Walsh-Transformation mit einer Einführung in die Bildverarbeitung, Vieweg, 1982.Google Scholar
  32. [32]
    R.A. Parker, The Computer Calculation of Modular Characters (The Meat-Axe), in: [1], 267–274.Google Scholar
  33. [33]
    W. Plesken, Towards a Soluble Quotient Algorithm,J. Symbolic Computation,4 (1987), 111–122.Google Scholar
  34. [34]
    D. Rockmore, Fast Fourier Analysis for Abelian Group Extensions,Advances in Applied Math.,11 (1990), 164–204.Google Scholar
  35. [35]
    A. Schönhage, V. Strassen, Schnelle Multiplikation großer Zahlen,Computing,7 (1971), 281–292.Google Scholar
  36. [36]
    J.P. Serre,Linear Representations of Finite Groups, Springer, 1977.Google Scholar
  37. [37]
    E.A. Trachtenberg, M.G. Karpovsky Filtering in a Communication Channel by Fourier Transforms over Finite Groups, in: [24], 179–216.Google Scholar

Copyright information

© Birkhäuser Verlag 1991

Authors and Affiliations

  • Ulrich Baum
    • 1
  1. 1.Institut für InformatikUniversität BonnBonnGERMANY

Personalised recommendations