Geometrical Interpretation of the Master Theorem for Divide-and-conquer Recurrences

TL;DR

This paper offers a geometric perspective on the Master Theorem for divide-and-conquer recurrences, linking solutions to fractal images, fractal dimensions, and Hausdorff measures, providing new insights into their structure.

Contribution

It introduces a novel geometric interpretation of the Master Theorem, connecting recurrence solutions with fractal geometry concepts.

Findings

Different recurrence cases correspond to distinct fractal images

Fractal dimension relates to the growth rate of recurrences

Hausdorff measure provides a measure for recurrence solutions

Abstract

We provide geometrical interpretation of the Master Theorem to solve divide-and-conquer recurrences. We show how different cases of the recurrences correspond to different kinds of fractal images. Fractal dimension and Hausdorff measure are shown to be closely related to the solution of such recurrences.