Computational Complexity of Biased Diffusion Limited Aggregation
Abstract
DiffusionLimited Aggregation (DLA) is a clustergrowth model that consists in a set of particles that are sequentially aggregated over a twodimensional 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 by $k$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 PComplete for 2DLA (thus for 3DLA). Later, we show that Prediction can be solved much more efficiently for 1DLA. In fact, we show that in that case the problem is NLComplete. With respect to Realization, we show that restricted to 2DLA the problem is in P, while in the 1DLA case, the problem is in L.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.10011
 Bibcode:
 2019arXiv190410011B
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Computational Complexity;
 Mathematics  Dynamical Systems;
 03D15;
 68Q17;
 68Q10