Computing Autotopism Groups of Partial Latin Rectangles

Abstract

Computing the autotopism group of a partial Latin rectangle (PLR) can be performed in multiple ways. This study has two aims: comparing some of these methods experimentally to identify those that are competitive; and identifying design goals for developing practical software. We compare six families of algorithms (two backtracking and four graph-theoretic methods), with and without using entry invariants (EIs), in a range of settings. Two EIs are considered: frequencies of row, column, and symbol representatives; and 2 × 2 submatrices. The best approach to computing autotopism groups varies. When PLRs have many autotopisms (such as having very few entries or being a group table), the McKay, Meynert, and Myrvold (MMM) method computes generators for the autotopism group efficiently. (The MMM method is the standard way to compute autotopisms.) Otherwise, PLRs ordinarily have trivial or small autotopism groups, and the task is to verify this. The so-called PLR graph method is slightly more efficient in this setting than the MMM method (in some circumstances, around twice as fast). With an intermediate number of entries, the quick-to-compute strong EIs are effective at reducing the need for computation without introducing significant overhead. With a full or almost-full PLR, a more sophisticated EI is needed to reduce down-the-line computation. These results suggest a hybrid approach to computing autotopism groups: The software decides on suitable EIs based on the input; and the user chooses between the MMM or the PLR graph methods, depending on their dataset. This article expands the authors’ previous article Computing autotopism groups of PLRs: a pilot study .

Recognition and Analysis of Image Patterns Based on Latin Squares by Means of Computational Algebraic Geometry

“…Similar studies concerning this new equivalence relation are, therefore, required and established as further work. In this regard, the joint use of the Latin square isomorphism invariants recently introduced in [34,35] may be of particular interest. It is also interesting to illustrate the computational efficiency of these two approaches in case of dealing with partial Latin squares with empty cells, whose distribution into isomorphism classes is only known [17,18] for order n ≤ 6.…”

“…The study of new invariants concerning partial Latin square isomorphisms has turned out to be an optimal approach to reduce this computational cost [33][34][35]. This paper delves in particular into those invariants that are related to affine algebraic sets associated to the partial Latin squares under consideration.…”