The Computational Complexity of Sandpiles

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.