The computational complexity of generating random fractals

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.

Parallel dynamics and computational complexity of network growth models

“…Examples include the Eden model, invasion percolation, the restricted solidon-solid model [5], the Bak-Sneppen model [6] and in-ternal diffusion-limited aggregation [7] all of which can be simulated in parallel in polylogarithmic time. On the other hand, no polylog time algorithm is known for generating diffusion-limited aggregation clusters and there is evidence that only powerlaw speed-ups are possible using parallelism [8,9].…”

Complexity, parallel computation and statistical physics

“…Is it possible that parallelism would permit many sweeps to be carried out in a much smaller number of parallel steps, thus reducing the bound on depth? The prospect for achieving reductions to polylog parallel time by compressing many Monte Carlo sweeps into a much smaller number of parallel steps is ruled out, modulo accepting P = NC, by the P-completeness proofs for natural decision problems associated with Metropolis and Swendsen-Wang dynamics [47]. A P-completeness result even holds for zero temperature single spin flip dynamics [48].…”

“…All of the decisions made during the construction of the pattern are independent and can be made simultaneously so there is evidently no history recorded in the final pattern. A Boolean circuit that generates Mandelbrot percolation patterns is straightforward to design [47]. Each square or pixel at the smallest scale is black only if it is not whited out at any level of the construction.…”

“…The mapping to a waiting time growth model and parallel solution using minimum weight paths can be applied to other cluster growth models with self-organized critical behavior [47], such as the Eden model [52,53] and the restricted solid-on-solid model [54,55]. For all of these examples, an apparently sequential growth process can be replaced by a parallel growth process that yields the same ensemble of cluster configurations in polylog parallel time.…”

Untitled

“…Several problems of interest in physics have been shown to be P-complete by Machta, Moore and others, including Ising models [18,23], diffusion-limited aggregation [16,18], lattice gases [22], cellular automata with local voting rules [23], and simple deterministic growth models [13]. Greenlaw et al [12] have pointed out that predicting a CA's evolution is P-complete in general, since CA rules exist (e.g.…”