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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{O}$$ \end{document}(n log n), and a parallel one that runs in time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{O}$$ \end{document}(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.