The computational complexity of sandpiles

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 C (n log n), and a parallel one that runs in time C (log3 n), 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.