Using fractal geometry for solving divide-and-conquer recurrences
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.
KeywordsFractals fractal dimension Hausdorff-Besikovitch dimension iterated function systems divide-and-conquer recurrences
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