Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity |
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Authors: | Moore Cristopher Machta Jonathan |
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Institution: | (1) Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico, 87501;(2) Computer Science Department and Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico, 87131;(3) Department of Physics and Astronomy, University of Massachusetts, Amherst, Massachusetts, 01003 |
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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
((n/k+logk)n
2/d
) steps. This is a significant speedup over explicit sequential simulation, which takes
(n
1+2/d
) time on average. Finally, we show that in one dimension internal DLA can be predicted in
(logn) parallel time, and so is in the complexity class NC. |
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Keywords: | internal diffusion-limited aggregation computational complexity parallel algorithms |
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