Efficient Computation Method for Identity Element of Abelian Sandpile Model

Abelian Sandpile Model (ASM) is one of the simple model for simulating the dynamics of sands by cellular automaton with a simple rule. ASM is defined on a finite graph and has certain algebraic structure. There exist a special configuration which is the identity element of this algebra. One problem is the time to compute the identity element. In this paper the authors propose the generalized burning method for ASM on undirected graphs and efficient algorithm for computing the identity element of the sandpile on the two-dimensional (n × n) finite lattice, rigorous computation time of which is conjectured to be O(n
³
log n). This method would be even available for other graphs. The algorithm deeply improves the computation time of the identity element since it takes O(n
⁴
log n) time to compute the identity element of ASM by the standard algorithm.