A CAT algorithm for the exhaustive generation of ice piles

Abstract

We present a CAT (constant amortized time) algorithm for generating those partitions of n that are in the ice pile model IPM k (n), a generalization of the sand pile model SPM(n). More precisely, for any fixed integer k, we show that the negative lexicographic ordering naturally identifies a tree structure on the lattice IPM k (n): this lets us design an algorithm which generates all the ice piles of IPM k (n) in amortized time O(1) and in space O(√ n).

An efficient algorithm for generating symmetric ice piles

“…In [10] it is shown how this function can be implemented so that if a = (n) then, for any fixed integer k, the iteration of the instruction a:=Next(a) (until M(a) = ∅) generates IPM k (n) = G((n)) in time O(|G((n))|) (and so is a CAT algorithm) using O( √ n) space. More generally, for any a ∈ IPM k (n) one can generate G(a) in time O(|G(a)|).…”

“…In particular, we show that SIPM k (n) can be generated by means of a CAT (Constant Amortized Time) algorithm. We recall that CAT algorithms for generating sand and ice piles have been presented in [9] and [10], and that a CAT algorithm for generating symmetric sand piles has been proposed in [11], where a decomposition property of symmetric sand piles in terms of product of sand piles is exploited. We prove that a similar decomposition property holds for symmetric ice piles: this lets us design a CAT algorithm that sequentially generates the elements of SIPM k (n) using O( √ kn) space.…”

On the exhaustive generation of plane partitions

“…On the other hand, together with some other researchers, the authors have been recently interested in the exhaustive generation [5] of the accessible configurations of a discrete dynamical system called the Ice Pile Model (IPM k (n)), describing the evolution of a granular environment (such as heaps of sand grains or snow flakes). Here, n is the number of ''grains" composing the system and k a parameter expressing the ability of the grains to slide on each other.…”

“…In this system, grains are stacked in adjacent columns (initially, they are all in the first column) and, under certain conditions, a grain on top of a column can be moved to the top of another column on the right. Such configurations can be represented by linear partitions of the integer n. The space of all accessible configurations for a fixed n gets larger when k increases and, when k is sufficiently large, it includes all linear partitions of n. Therefore, the algorithm presented in [5] provides an alternative CAT method to generate all linear partitions of an integer n.…”

“…We call this the horizontal projection of the plane partition. Therefore, the Massazza-Radicioni CAT algorithm [5] can be used to generate all possible horizontal projections. The problem then boils down to generating all plane partitions having a fixed linear partition π as horizontal projection.…”