  Diffusion-Limited Aggregation (DLA) is a cluster-growth model that consists
in 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 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 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 restricted to 2-DLA the problem is in P, while in the 1-DLA case, the
problem is in L.
