The Complexity of Three-Dimensional Critical Avalanches

In this work we study the complexity of the three-dimensional sandpile avalanches triggered by the addition of two critical con…gurations. We prove that the algorithmic problem consisting in predicting the evolution of three dimensional critical avalanches is P -complete and 6 p n-strict hard for P . On the other hand we prove that three-dimensional avalanches are superlinear long on average. It suggests that the prediction problem is superlinear-hard on average. Can we quickly predict the evolution of an avalanche if we are given a full description of the initial conditions? The Abelian Sandpile Model has been used to simulate dissipative dynamical process such as forest …res, earth quakes, extinction events, and (o¤ course) avalanches [3]. Can we quickly predict sandpile avalanches? There is some previous work concerning the computational complexity of prediction problems related to The Abelian Sandpile Model (see for example [4], and [5]). Most of those works are focused on the analysis of The Sandpile Prediction Problem, which refers to the computation of relaxations of unstable con…gurations. In this work we analyze the complexity of predicting the …nal state of the avalanches triggered by the addition of two critical con…gurations, (we focus our research on three-dimensional cubic lattices). Those avalanches are called critical avalanches. We show that GC; the problem consisting in predicting the evolution of three-dimensional critical avalanches, is at least as hard as most of the algorithmic problems related to The Abelian Sandpile Model, that is: we show that GC is the hardness core of the predicting tasks related to the model. It is important to remark that our complexity theoretical analysis is based on the notion of NC-Turing reducibility. We have chosen to work with this notion because all the algorithmic problems considered in this paper are Ptime computable, and because we are interested in analyzing the polylogarithmic time computability of those problems. We believe that the argued Self-organized Criticality [1] of The Abelian Sandpile Model is the complexity source of GC and its relatives. We show that GC is P -complete; and we prove that critical avalanches are superlinear long on average. It suggests that any sequential simulation algorithm computing GC has a running time which is superlinear on average. Also, we prove that the criticality of the model implies some type of average-case hardness. We wanted to establish some links between the Self-Organized Criticality of The Abelian Sandpile