Using fractal geometry for solving divide-and-conquer recurrences

  • Simant Dube
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 762)


A relationship between the fractal geometry and the analysis of recursive (divide-and-conquer) algorithms is investigated. It is shown that the dynamic structure of a recursive algorithm which might call other algorithms in a mutually recursive fashion can be geometrically captured as a fractal (self-similar) image. This fractal image is defined as the attractor of a mutually recursive function system. It then turns out that the Hausdorff-Besikovitch dimension D of such an image is precisely the exponent in the time complexity of the algorithm being modeled. That is, if Hausdorff D-dimensional measure of the image is finite then it serves as the constant of proportionality and the time complexity is of the form gQ(nD), else it implies that the time complexity is of the form θ(nD logpn) where p is an easily determined constant.


Fractals fractal dimension Hausdorff-Besikovitch dimension iterated function systems divide-and-conquer recurrences 


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  1. [1]
    A. V. Aho, J. E. Hopcroft and J. D. Ullman, The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974.Google Scholar
  2. [2]
    M. F. Barnsley, Fractals Everywhere, Academic Press, 1988.Google Scholar
  3. [3]
    M. F. Barnsley, J. H. Elton and D. P. Hardin, “Recurrent Iterated Function Systems,” Constructive Approximation, 5, 3–31 (1989).CrossRefGoogle Scholar
  4. [4]
    M. F. Barnsley, R. L. Devaney, B. B. Mandelbrot, H-O. Peitgen, De Saupe, and R. F. Voss, Science of Fractal Images, Springer-Verlag, 1988.Google Scholar
  5. [5]
    J. L. Bentley, D. Haken and J. B. Saxe, “A General Method for Solving Divideand-Conquer Recurrences,” SIGACT News, 12, 36–44 (1980).CrossRefGoogle Scholar
  6. [6]
    T. H. Cormen, C. E. Leiserson and R. L. Rivest, Introduction to Algorithms, MIT Press, 1990.Google Scholar
  7. [7]
    K. Culik II and S. Dube, “Affine Automata and Related Techniques for Generation of Complex Images,” Theoretical Computer Science 116, 373–398 (1993).CrossRefGoogle Scholar
  8. [8]
    K. Culik II and S. Dube, “Rational and Affine Expressions for Image Synthesis.” Discrete Applied Mathematics 41, 85–120 (1993).CrossRefGoogle Scholar
  9. [9]
    K. Culik II and S. Dube, “Balancing Order and Chaos in Image Generation,” Preliminary version in Proc. of ICALP'91. Lecture notes in Computer Science 510, Springer-Verlag, pp. 600–614. Computer and Graphics, to appear.Google Scholar
  10. [10]
    S. Dube, “Using Fractal Geometry for Solving Divide-and-Conquer Recurrences,” Technical Report 93-70, Dept of Math, Stat and Comp Sci, University of New England at Armidale, Australia.Google Scholar
  11. [11]
    R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989.Google Scholar
  12. [12]
    D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms, Birkhäuser Boston, 1982.Google Scholar
  13. [13]
    B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., San Francisco, 1982.Google Scholar
  14. [14]
    R. D. Mauldin and S. C. Williams, “Hausdorff Dimension in Graph Directed Constructions,” Transactions of American Mathematical Society, 309, 811–829 (1988).Google Scholar
  15. [15]
    P. W. Purdom, Jr. and C. A. Brown, The Analysis of Algorithms, Holt, Rinehart and Winston, 1985.Google Scholar
  16. [16]
    L. Staiger, “Quadtrees and the Hausdorff Dimension of Pictures,” Workshop on Geometrical Problems of Image Processing (GEOBILD'89), Math. Research No. 51, Akademie-Verlag, Berlin, pp. 173–178 (1989).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Simant Dube
    • 1
  1. 1.Department of Mathematics, Statistics and Computing ScienceUniversity of New EnglandArmidaleAustralia

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