Journal of Statistical Physics

, Volume 99, Issue 3–4, pp 661–690

Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity

  • Cristopher Moore
  • Jonathan Machta
Article

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)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 aggregation computational complexity parallel algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    F. Barahona, On the computational complexity of Ising spin glass models, J. Phys. A 15:3241–3253 (1982).Google Scholar
  2. 2.
    R. M. Brady and R. C. Ball, Fractal growth of copper electrodeposits, Nature 309:225–229 (1984).Google Scholar
  3. 3.
    M. Bramson and J. L. Lebowitz, Asymptotic behavior of densities for two-particle annihilating random walks, J. Stat. Phys. 62:297–372 (1991).Google Scholar
  4. 4.
    A. Condon, A theory of strict P-completeness, STACS 1992, in Lecture Notes in Computer Science 577:33–44 (1992).Google Scholar
  5. 5.
    D. Dhar, The Abelian sandpile and related models, Physica A 263:4–25 (1999).Google Scholar
  6. 6.
    P. Diaconis and W. Fulton, A growth model, a game, an algebra, Lagrange inversion, and characteristic classes, Rend. Sem. Mat. Univ. Pol. Torino 49:95–119 (1991).Google Scholar
  7. 7.
    R. M. D'souza and N. H. Margolus, A thermodynamically reversible generalization of diffusion-limited aggregation, Phys. Rev. E. 60:264–274 (1999).Google Scholar
  8. 8.
    F. Family, B. R. Masters, and D. E. Platt, Fractal pattern formation in human retinal vessels, Physica D 38:98–103 (1989).Google Scholar
  9. 9.
    J. Gravner and J. Quastel, Internal DLA and the Stefan problem, Ann. Prob., to appear.Google Scholar
  10. 10.
    R. Greenlaw, H. J. Hoover, and W. L. Ruzzo, Limits to Parallel Computation: P-Com-pleteness Theory (Oxford University Press, 1995).Google Scholar
  11. 11.
    D. Griffeath and C. Moore, Life without death is P-complete, Complex Systems 10: 437–447 (1996).Google Scholar
  12. 12.
    H. G. E. Hentschel and A. Fine, Diffusion-regulated control of cellular dendritic morphogenesis, Proc. R. Soc. Lond. B 263:1–8 (1996).Google Scholar
  13. 13.
    D. E. Knuth, Seminumerical Algorithms (Addison-Wesley, 1981).Google Scholar
  14. 14.
    J. Krug and P. Meakin, Kinetic roughening of Laplacian fronts, Phys. Rev. Lett. 66:703 (1991).Google Scholar
  15. 15.
    H. Larralde, P. Trunfio, S. Havlin, H. E. Stanley, and G. H. Weiss, Territory covered by N diffusing particles, Nature 355:423–426 (1992).Google Scholar
  16. 16.
    G. Lawler, M. Bramson, and D. Griffeath, Internal diffusion limited aggregation, Ann. Prob. 20:2117-2140 (1992).Google Scholar
  17. 17.
    G. Lawler, Subdiffusive fluctuations for internal diffusion limited aggregation, Ann. Prob. 23:71–86 (1995).Google Scholar
  18. 18.
    K. Lindgren and M. G. Nordahl, Universal computation in simple one-dimensional cellular automata, Complex Systems 4:299–318 (1990).Google Scholar
  19. 19.
    J. Machta, The computational complexity of pattern formation, J. Stat. Phys. 70:949 (1993).Google Scholar
  20. 20.
    J. Machta and R. Greenlaw, The parallel complexity of growth models, J. Stat. Phys. 77:755 (1994).Google Scholar
  21. 21.
    J. Machta and R. Greenlaw, The computational complexity of generating random fractals, J. Stat. Phys. 82:1299 (1996).Google Scholar
  22. 22.
    J. G. Masek and D. L. Turcotte, A diffusion-limited aggregation model for the evolution of drainage networks, Earth and Planetary Science Letters 119:379–386 (1993).Google Scholar
  23. 23.
    E. W. Mayr and A. Subramanian, The complexity of circuit value and network stability, J. Comput. System Sci. 44:302–323 (1992).Google Scholar
  24. 24.
    P. Meakin and J. M. Deutch, The formation of surfaces by diffusion-limited annihilation, J. Chem. Phys. 85:2320 (1986).Google Scholar
  25. 25.
    A. M. Meirmanov, The Stefan Problem (Walter de Gruyter, Berlin, 1992).Google Scholar
  26. 26.
    C. Moore and M. Nilsson, The computational complexity of sandpiles, J. Stat. Phys. 96:205–224 (1999).Google Scholar
  27. 27.
    C. Moore, Majority-Vote cellular automata, Ising dynamics, and P-completeness, J. Stat. Phys. 88:795–805 (1997).Google Scholar
  28. 28.
    C. Moore and M. Nordahl, Predicting lattice gases is P-complete (Santa Fe Institute Working Paper 97-04-034).Google Scholar
  29. 29.
    C. Moore, Quasi-linear cellular automata, Physica D 103:100–132 (1997); Proceedings of the International Workshop on Lattice Dynamics.Google Scholar
  30. 30.
    C. Moore, Predicting non-linear cellular automata quickly by decomposing them into linear ones, Physica D 111:27–41 (1998).Google Scholar
  31. 31.
    K. Moriarty and J. Machta, The computational complexity of the Lorentz lattice gas, J. Stat. Phys. 87:1245 (1997).Google Scholar
  32. 32.
    K. Moriarty, J. Machta, and R. Greenlaw, Optimized parallel algorithm and dynamic exponent for diffusion-limited aggregation, Phys. Rev. E 55:6211 (1997).Google Scholar
  33. 33.
    L. Niemeyer, L. Pietronero, and H. J. Wiesmann, Fractal dimension of dielectric breakdown, Phys. Rev. Lett 52:1033–1036 (1984).Google Scholar
  34. 34.
    J. Nittmann and H. E. Stanley, Tip splitting without interfacial tension and dendritic growth patterns arising from molecular anisotropy, Nature 321:663 (1986).Google Scholar
  35. 35.
    J. Nittmann and H. E. Stanley, Non-deterministic approach to anisotropic growth pat-terns with continuously tunable morphology: the fractal properties of some real snow-flakes, J. Phys. A 20: L1185 (1987).Google Scholar
  36. 36.
    C. H. Papadimitriou, Computational Complexity (Addison-Wesley, 1994).Google Scholar
  37. 37.
    L. Paterson, Diffusion-limited aggregation and two-fluid displacement in porous media, Phys. Rev. Lett. 52:1621 (1984).Google Scholar
  38. 38.
    C. Tang, Diffusion-limited aggregation and the Saffman-Taylor problem, Phys. Rev. A 31: 1977 (1985).Google Scholar
  39. 39.
    D. Toussaint and F. Wilczek, Particle-antiparticle annihilation in diffusive motion, J. Chem. Phys. 78:2642–2647 (1983).Google Scholar
  40. 40.
    T. Witten and L. Sander, Diffusion-limited aggregation: a kinetic critical phenomenon, Phys. Rev. Lett. 47:1400-1403 (1981).Google Scholar

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

Personalised recommendations