The Parallel Complexity of Growth Models

TL;DR

This paper develops fast parallel algorithms for simulating various growth models, revealing their lower complexity compared to more intricate models like diffusion-limited aggregation.

Contribution

It introduces novel parallel randomized algorithms with polylogarithmic running time for key growth models, linking their complexity to self-avoiding paths in random environments.

Findings

Algorithms run in O(log^2 N) time

Growth models are less complex than DLA

Parallel algorithms enable efficient simulation

Abstract

This paper investigates the parallel complexity of several non-equilibrium growth models. Invasion percolation, Eden growth, ballistic deposition and solid-on-solid growth are all seemingly highly sequential processes that yield self-similar or self-affine random clusters. Nonetheless, we present fast parallel randomized algorithms for generating these clusters. The running times of the algorithms scale as $O(\log^2 N)$, where $N$ is the system size, and the number of processors required scale as a polynomial in $N$. The algorithms are based on fast parallel procedures for finding minimum weight paths; they illuminate the close connection between growth models and self-avoiding paths in random environments. In addition to their potential practical value, our algorithms serve to classify these growth models as less complex than other growth models, such as diffusion-limited aggregation, for which fast parallel algorithms probably do not exist.