Computational Complexity of Biased Diffusion-Limited Aggregation

Abstract

Diffusion-Limited Aggregation (DLA) is a cluster growth model that consists of a set of particles that are sequentially aggregated over a two-dimensional grid. In this paper, we introduce a biased version of the DLA model, in which particles are limited to move in a subset of possible directions. We denote $k$-DLA the model where the particles move only in $k$ possible directions. We study the biased DLA model from the perspective of Computational Complexity, defining two decision problems The first problem is Prediction, whose input is a site of the grid $c$ and a sequence $S$ of walks, representing the trajectories of a set of particles. The question is whether a particle stops at site $c$ when sequence $S$ is realized. The second problem is Realization, where the input is a set of positions of the grid, $P$. The question is whether there exists a sequence $S$ that realizes $P$, i.e. all particles of $S$ exactly occupy the positions in $P$. Our aim is to classify the Prediciton and Realization problems for the different versions of DLA. We first show that Prediction is P-Complete for 2-DLA (thus for 3-DLA). Later, we show that Prediction can be solved much more efficiently for 1DLA. In fact, we show that in that case, the problem is NL-Complete. With respect to Realization, we show that restricted to 2DLA the problem is in P, while for 1DLA is in L.