Abstract
In this interdisciplinary research we apply the tools of algorithmic complexity theory to three important non-equilibrium growth models that are used in statistical physics. Much of the insight into these models has been derived by numerical simulation; it is important to develop good parallel algorithms for them. Eden growth, solidon-solid growth and ballistic deposition are all seemingly highly sequential processes. However, we are able to provide algorithms for the models that run in time O(log2 N) using a polynomial number of processors on a randomized CREW P-RAM, where N is the system size. In addition to their potential practical value, our algorithms serve to classify these growth models as less complex than other growth models, such as diffusion-limited aggregation, for which fast parallel algorithms probably do not exist.
This research was partially funded by the National Science Foundation Grant CCR-9209184.
This research was partially funded by the National Science Foundation Grant DMR-9311580.
Preview
Unable to display preview. Download preview PDF.
References
M. Eden. A two-dimensional growth process. In F. Neyman, editor, Proceeding of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, volume IV, page 223. University of California, 1961.
F. Family and T. Vicsek, editors. Dynamics of Fractal Surfaces. World Scientific, Singapore, 1991.
F. E. Fich. The complexity of computation on the parallel random access machine. In J. H. Reif, editor, Synthesis of Parallel Algorithms, chapter 20, pages 843–899. Morgan Kaufman, San Mateo, CA, 1993.
P. Freche, D. Stauffer, and H. E. Stanley. Surface structure and anisotropy of Eden clusters. J. Phys. A: Math. Gen., 18:L1163, 1985.
A. Gibbons and W. Rytter. Efficient Parallel Algorithms. Cambridge University Press, 1988.
R. Greenlaw, H. J. Hoover, and W. L. Ruzzo. Topics in Parallel Computation: A Guide to P-completeness Theory. Computing Science Series, editor Z. Galil. Oxford University Press, to appear.
M. Kardar, G. Parisi, and Y.-C. Zheng. Dynamic scaling of growing interfaces. Phys. Rev. Lett., 56:889, 1986.
R. M. Karp and V. Ramachandran. Parallel algorithms for shared-memory machines. In Jan van Leeuwan, editor, Handbook of Theoretical Computer Science, volume A: Algorithms and Complexity, chapter 17, pages 869–941. M.I.T. Press/Elsevier, 1990.
H. Kesten. Percolation Theory for Mathematicians. Birkhauser, Boston, 1982.
J. M. Kim and J. M. Kosterlitz. Growth in a restricted solid-on-solid model. Phys. Rev. Lett., 62:2289, 1989.
J. Krug and H. Spohn. Kinetic roughening of growing surfaces. In C. Godreche, editor, Solids Far From Equilibrium: Growth, Morphology and Defects. Cambridge University Press, 1991.
J. Machta. The computational complexity of pattern formation. J. Stat. Phys., 70:949, 1993.
J. Machta and R. Greenlaw. P-complete problems in statistical physics. Unpublished, 1994.
P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball. Ballistic deposition on surfaces. Phys. Rev. A, 34:5091, 1986.
M. Plischke and Z. Racz. Active zone of growing clusters: Diffusion-limited aggregation and the Eden model. Phys. Rev. Lett., 53:415, 1984.
D. Richardson. Random growth in a tessellation. Proc. Camb. Phil. Soc., 74:515, 1973.
S. Roux, A. Hansen, and E. L. Hinrichsen. A direct mapping between Eden growth model and directed polymers in random media. J. Phys. A: Math. Gen., 24:L295, 1991.
L.-H. Tang, J. Kertesz, and D. E. Wolf. Kinetic roughening with power-law waiting time distribution. J. Phys. A: Math. Gen., 24:L1193, 1991.
T. Vicsek. Fractal Growth Phenomena. World Scientific, Singapore, 1992.
M. J. Vold. A numerical approach to the problem of sediment volume. J. Colloid Sci., 14:168, 1959.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Greenlaw, R., Machta, J. (1994). The parallel complexity of eden growth, solid-on-solid growth and ballistic deposition. In: van Leeuwen, J. (eds) Algorithms — ESA '94. ESA 1994. Lecture Notes in Computer Science, vol 855. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0049429
Download citation
DOI: https://doi.org/10.1007/BFb0049429
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58434-6
Online ISBN: 978-3-540-48794-4
eBook Packages: Springer Book Archive