# Algorithmic interpretations of fractal dimension

## TL;DR

This paper introduces an algorithmic framework for sets with low fractal dimension in Euclidean space, leading to faster algorithms for classical problems, with performance nearly matching that based on ambient dimension.

## Contribution

It defines a fractal dimension for finite metric spaces that aligns with empirical notions and leverages it to improve algorithmic efficiency in geometric problems.

## Key findings

- Faster algorithms for classical geometric problems on low fractal dimension sets.
- Performance dependence on fractal dimension closely matches that on Euclidean dimension.
- Results apply to exact, fixed-parameter algorithms, approximation schemes, and spanners.

## Abstract

We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several well-studied notions of fractal dimension for sets and measures in Euclidean space. We consider a definition of fractal dimension for finite metric spaces which agrees with standard notions used to empirically estimate the fractal dimension of various sets. We define the fractal dimension of some metric space to be the infimum $\delta>0$, such that for any $\epsilon > 0$, for any ball $B$ of radius $r\geq 2\epsilon$, and for any $\epsilon $-net $N$ (that is, for any maximal $\epsilon $-packing), we have $|B\cap N|=O((r/\epsilon)^\delta)$. Using this definition we obtain faster algorithms for a plethora of classical problems on sets of low fractal dimension in Euclidean space. Our results apply to exact and fixed-parameter algorithms, approximation schemes, and spanner constructions. Interestingly, the dependence of the performance of these algorithms on the fractal dimension nearly matches the currently best-known dependence on the standard Euclidean dimension. Thus, when the fractal dimension is strictly smaller than the ambient dimension, our results yield improved solutions in all of these settings.