Journal of Statistical Physics

, Volume 96, Issue 1, pp 205–224

The Computational Complexity of Sandpiles

  • Cristopher Moore
  • Martin Nilsson

DOI: 10.1023/A:1004524500416

Cite this article as:
Moore, C. & Nilsson, M. Journal of Statistical Physics (1999) 96: 205. doi:10.1023/A:1004524500416


Given an initial distribution of sand in an Abelian sandpile, what final state does it relax to after all possible avalanches have taken place? In d≥3, we show that this problem is P-complete, so that explicit simulation of the system is almost certainly necessary. We also show that the problem of determining whether a sandpile state is recurrent is P-complete in d≥3, and briefly discuss the problem of constructing the identity. In d=1, we give two algorithms for predicting the sandpile on a lattice of size n, both faster than explicit simulation: a serial one that runs in time \(\mathcal{O}\)(n log n), and a parallel one that runs in time \(\mathcal{O}\)(log3n), i.e., the class NC3. The latter is based on a more general problem we call additive ranked generability. This leaves the two-dimensional case as an interesting open problem.

sandpiles self-organized criticality cellular automata computational complexity parallel computation nonlinear systems Boolean circuits graph theory 

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • Cristopher Moore
    • 1
  • Martin Nilsson
    • 2
  1. 1.Santa Fe InstituteSanta Fe
  2. 2.Chalmers Tekniska Högskola and University of GothenburgGöteborg

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