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Using fractal geometry for solving divide-and-conquer recurrences

Published online by Cambridge University Press:  17 February 2009

Simant Dube
Simant Dube
Affiliation:
Iterated Systems Inc., Seven Piedmont Centre, Atlanta GA 30305, USA.
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Abstract

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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-Besicovich dimension D of such an image is precisely the exponent in the time complexity of the algorithm being modelled. That is, if the 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 Θ(nD), else it implies that the time complexity is of the form Θ(nD logpn), where p is an easily determined constant.

Type
Research Article
Information
The ANZIAM Journal , Volume 37 , Issue 2 , October 1995 , pp. 145 - 171
Copyright
Copyright © Australian Mathematical Society 1995

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