# The Parallel Simulation Method for d-dimensional Abelian Sandpile Automata

## Keywords:

Abelian sandpile model, Dense packing problem, Parallel algorithm, Cellular automata## Abstract

In this paper, the star-packing problem introduced in [1] for a square lattice is generalized for d-dimensional lattice Ld, d ∈ N. The problem is to pack the lattice Ld with star graphs S2d. Using the solution of this problem, a parallel algorithm for the simulation of d-dimensional cellular automata is developed. As an example of cellular automata, the relaxation process of unstable states of Abelian sandpile model is considered. Appropriate software packages have been developed using OpenMP and CUDA technologies. The parallel simulation results, carried out for 3-dimensional lattices of different sizes, are presented.

## References

V. S. Poghosyan, S. S. Poghosyan and H. E. Nahapetyan, “The investigation of models of self-organized systems by parallel programming methods based on the example of an abelian sandpile model”, Proc. CSIT Conference 2013, Yerevan Armenia, Sept. 23-27, pp. 260–262, 2013.

E. Davtyan, H. Karapetyan and K. Shahbazyan, “Software tool for cluster-based modeling of 2d cellular automata”, Proc. CSIT Conferance, Yerevan Armenia, 28 Sept. - 2 Oct., pp. 404–407, 2013.

S. Frehmel, “The sandpile model: parallelization of efficient algorithms for systems with shared memory”, ACRI, Lecture Notes in Computer Science 6350, pp. 35–45, 2010.

P. Bak, C. Tang and K. Wiesenfeld, “Self-organized criticality: An explanation of the 1/f noise”, Phys. Rev. Lett., vol.59, no. 4, pp. 381–384, 1987.

D. Dhar, “Self-organized critical state of sandpile automaton models”, Phys. Rev. Lett., vol. 64, no. 14, pp. 1613–1616, 1990.

D. Dhar, “Theoretical studies of self-organized criticality”, Physica A, vol. 369, no. 1, pp. 29–70, 2006.

P. Grassberger and S. S. Manna, “Some more sandpiles”, J. Phys. France, vol. 51, pp. 1077–1098, 1990.

V. S. Poghosyan, S. Y. Grigorev, V. B. Priezzhev and P. Ruelle, “Pair correlations in the sandpile model: A check of logarithmic conformal field theory”, Phys. Lett. B, vol. 659, pp. 768–772, 2008.

Su. S. Poghosyan, V. S. Poghosyan, V. B. Priezzhev and P. Ruelle, “Numerical study of correspondence between the dissipative and fixed-energy Abelian sandpile models”, Phys.Rev. E, 84, 066119, 2011.

A. Fey, L. Levine, and D.B. Wilson, “Driving Sandpiles to Criticality and Beyond”, Phys. Rev. Lett., 104, 145703, 2010.

A. Fey, L. Levine, and D. B. Wilson, “Approach to criticality in sandpiles”, Phys. Rev. E 82, 031121 (2010).

V. S. Poghosyan, S. Y. Grigorev, V. B. Priezzhev and P. Ruelle, “Logarithmic two-point correlators in the Abelian sandpile model”, J. Stat. Mech., vol. 2010, no. 07, P07025, 2010.

V. S. Poghosyan and V. B. Priezzhev, “The problem of predecessors on spanning trees”, Acta Polytechnica, vol. 51, no. 1, pp. 59–62, 2011.

S. N. Majumdar and D. Dhar, “Height correlations in the Abelian sandpile model”, J. Phys. A: Math. Gen., vol. 24, no. 7, L357–L362, 1991.

O.Delmas, S. Perennes, “Circuit-switched gossiping in 3-dimensional torus networks”, Proc. Euro-Par, Parallel-Processing, pp. 370–373, 1996.

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*Mathematical Problems of Computer Science*,

*46*, 117–125. Retrieved from http://mpcs.sci.am/index.php/mpcs/article/view/157

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