Computational Complexity of Avalanches in the Kadanoff Sandpile Model

Authors:
Enrico Formenti;Eric Goles;Bruno Martin
Affiliations:
(Correspd.) (Work supported by The French National Research Agency project EMC (ANR-09-BLAN-0164)) Université Nice-Sophia Antipolis, F-06903 Sophia Antipolis Cedex, France. enrico.formenti@un ...;(FONDECYT 1100003, BASAL-CMM-U. de Chile, Anillo ACT-88, CNRS) Universidad Adolfo Ibañez, Av. Diagonal Las Torres 2640, Peñalolen, Santiago, Chile. eric.chacc@uai.cl;(Work supported by The French National Research Agency project EMC (ANR-09-BLAN-0164)) Université Nice-Sophia Antipolis, F-06903 Sophia Antipolis Cedex, France. Bruno.Martin@i3s.unice.fr
Venue:
Fundamenta Informaticae - From Physics to Computer Science: to Gianpiero Cattaneo for his 70th birthday
Year:
2012

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Abstract

This paper investigates the avalanche problem AP for the Kadanoff sandpile model (KSPM). We prove that (a slight restriction of) AP is in NC1 in dimension one, leaving the general case open. Moreover, we prove that AP is P-complete in dimension two. The proof of this latter result is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with an initial sand distribution in KSPM. These results are also related to the known prediction problem for sandpiles which is in NC1 for one-dimensional sandpiles and P-complete for dimension 3 or higher. The computational complexity of the prediction problem remains open for the Bak's model of two-dimensional sandpiles.