The Computational Complexity of Sandpiles

TL;DR

This paper investigates the computational complexity of sandpile models, proving P-completeness in three or more dimensions and providing efficient algorithms for one-dimensional cases, leaving the two-dimensional case open.

Contribution

It establishes P-completeness for sandpile relaxation and recurrence in higher dimensions and introduces faster algorithms for 1D sandpile prediction.

Findings

Relaxation problem is P-complete in d >= 3

Recurrence determination is P-complete in d >= 3

Efficient algorithms for 1D sandpile prediction

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. 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 O(n log n), and a parallel one that runs in time O(log^3 n), i.e. in 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.