Computational Complexity of Restricted Diffusion Limited Aggregation
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We introduce a restricted version of the Diffusion Limited Aggregation (DLA) model. DLA is a cluster growth model that consists in series of particles that are thrown one by one from the top edge of a two (on more) dimensional grid, where they undergo a random walk until they join the cluster. In our restricted version, particles are limited to move in a subset of the possible directions. We study the restricted DLA model in the context of Computational Complexity, defining a decision problem consisting in computing whether a given cell in the grid will be occupied after the deterministic dynamics have taken place. We find that depending on the imposed restrictions, this problem can be either efficiently parallelizable of inherently sequential. More precisely, we show that the prediction problem associated with particles allowing two or more movement directions is P-Complete, as it can simulate arbitrary Boolean circuits. On the hand, the prediction problem associated to particles restricted to move in one direction is much more limited, despite still having non-trivial computational capabilities. Indeed, in this case we show that the prediction problem can be solved by a fast parallel algorithm.
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