Computational complexity of the avalanche problem on one dimensional Kadanoff sandpiles

TL;DR

This paper proves that the avalanche problem for one-dimensional Kadanoff sandpiles is in NC, indicating it can be efficiently solved in parallel, contrasting with higher-dimensional cases that are P-complete.

Contribution

It establishes that the avalanche problem for 1D Kadanoff sandpiles is in NC, resolving an open problem and highlighting dimension-sensitive computational complexity.

Findings

The avalanche problem in 1D Kadanoff sandpiles is in NC.

Higher dimensions make the problem P-complete.

The result differentiates complexity based on dimension.

Abstract

In this paper we prove that the general avalanche problem AP is in NC, for the Kadanoff sandpile model in one dimension, answering an open problem of Formenti, Goles and Martin in 2010. Thus adding one more item to the (slowly) growing list of dimension sensitive problems since in higher dimensions the problem is P-complete (for monotone sandpiles).