Computational Complexity of Biased Diffusion-Limited Aggregation
Nicolás Bitar, Eric Goles, Pedro Montealegre · SIAM Journal on Discrete Mathematics · 2022
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 moving 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 Prediction 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 1-DLA. In fact, we show that in that case, the problem is NL-Complete. With respect to Realization, we show that when restricted to 2-DLA the problem is in P, while in the 1-DLA case, the problem is in L.