# Computational Complexity of Biased Diffusion Limited Aggregation

## TL;DR

This paper investigates the computational complexity of a biased diffusion-limited aggregation model, classifying the difficulty of predicting particle placement and realization problems across different directional constraints.

## Contribution

It introduces a biased DLA model with limited movement directions and classifies the complexity of associated decision problems, revealing P-Completeness and NL-Completeness results.

## Key findings

- Prediction is P-Complete for 2-DLA.
- Prediction is NL-Complete for 1-DLA.
- Realization problem is in P for 2-DLA and in L for 1-DLA.

## Abstract

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.