Journal of Statistical Physics

, Volume 99, Issue 3, pp 661–690

Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity

  • Cristopher Moore
  • Jonathan Machta

DOI: 10.1023/A:1018627008925

Cite this article as:
Moore, C. & Machta, J. Journal of Statistical Physics (2000) 99: 661. doi:10.1023/A:1018627008925


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)n2/d) steps. This is a significant speedup over explicit sequential simulation, which takes \(\mathcal{O}\)(n1+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.

internal diffusion-limited aggregationcomputational complexityparallel algorithms

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Cristopher Moore
    • 1
    • 2
  • Jonathan Machta
    • 3
  1. 1.Santa Fe InstituteSanta FeNew Mexico
  2. 2.Computer Science Department and Department of Physics and AstronomyUniversity of New MexicoAlbuquerqueNew Mexico
  3. 3.Department of Physics and AstronomyUniversity of MassachusettsAmherst