Abstract
The computational complexity of internal diffusion-limited aggregation (DLA) is examined from both a theoretical and a practical point of view. We show that for two or more dimensions, the problem of predicting the cluster from a given set of paths is complete for the complexity class CC, the subset of P characterized by circuits composed of comparator gates. CC-completeness is believed to imply that, in the worst case, growing a cluster of size n requires polynomial time in n even on a parallel computer. A parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths, and then corrects defects in the guessed cluster through a nonlocal annihilation process. The parallel running time of the relaxation algorithm for two-dimensional internal DLA is studied by simulating it on a serial computer. The numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n. Thus the computational resources needed to grow large clusters are significantly less on average than the worst-case analysis would suggest. For a parallel machine with k processors, we show that random clusters in d dimensions can be generated in \(\mathcal{O}\)((n/k+logk)n 2/d) steps. This is a significant speedup over explicit sequential simulation, which takes \(\mathcal{O}\)(n 1+2/d) time on average. Finally, we show that in one dimension internal DLA can be predicted in \(\mathcal{O}\)(logn) parallel time, and so is in the complexity class NC.
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Moore, C., Machta, J. Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity. Journal of Statistical Physics 99, 661–690 (2000). https://doi.org/10.1023/A:1018627008925
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DOI: https://doi.org/10.1023/A:1018627008925