Internal DiffusionLimited Aggregation: Parallel Algorithms and Complexity
 Cristopher Moore,
 Jonathan Machta
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The computational complexity of internal diffusionlimited 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. CCcompleteness 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 twodimensional 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 worstcase 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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 Title
 Internal DiffusionLimited Aggregation: Parallel Algorithms and Complexity
 Journal

Journal of Statistical Physics
Volume 99, Issue 34 , pp 661690
 Cover Date
 20000501
 DOI
 10.1023/A:1018627008925
 Print ISSN
 00224715
 Online ISSN
 15729613
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 internal diffusionlimited aggregation
 computational complexity
 parallel algorithms
 Industry Sectors
 Authors

 Cristopher Moore ^{(1)} ^{(2)}
 Jonathan Machta ^{(3)}
 Author Affiliations

 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