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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 \bfP - 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 \bfN \bfL -Complete. With respect to Realization, we show that when restricted to 2-DLA the problem is in \bfP, while in the 1-DLA case, the problem is in \bfL .

Original languageEnglish
Pages (from-to)823-866
Number of pages44
JournalSIAM Journal on Discrete Mathematics
Issue number1
StatePublished - 2022
Externally publishedYes


  • NL-completeness
  • P-completeness
  • computational complexity
  • diffusion-limited aggregation
  • space complexity


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