The Computational Complexity of Generating Random Fractals

TL;DR

This paper analyzes the computational complexity of generating various random fractals, revealing some models are highly sequential while others can be simulated efficiently in parallel, impacting understanding of their intrinsic computational difficulty.

Contribution

It provides a complexity-theoretic comparison of different random fractal models, highlighting which can be efficiently parallelized and which are inherently sequential.

Findings

Diffusion limited aggregation is highly sequential and unlikely to be parallelized efficiently.

Mandelbrot percolation can be simulated in constant parallel time.

The study clarifies the intrinsic complexity differences among fractal models.

Abstract

In this paper we examine a number of models that generate random fractals. The models are studied using the tools of computational complexity theory from the perspective of parallel computation. Diffusion limited aggregation and several widely used algorithms for equilibrating the Ising model are shown to be highly sequential; it is unlikely they can be simulated efficiently in parallel. This is in contrast to Mandelbrot percolation that can be simulated in constant parallel time. Our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory. In addition, the results may serve as a guide to simulation physics.