Abstract
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 \(\mathcal{O}\)(log3 n), i.e., the class NC 3. 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.
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Moore, C., Nilsson, M. The Computational Complexity of Sandpiles. Journal of Statistical Physics 96, 205–224 (1999). https://doi.org/10.1023/A:1004524500416
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DOI: https://doi.org/10.1023/A:1004524500416