# Using fractal geometry for solving divide-and-conquer recurrences

## Abstract

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(*n*^{D}), else it implies that the time complexity is of the form θ(*n*^{D} log^{p}*n*) where *p* is an easily determined constant.

## Keywords

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

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