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View all- Falcón R(2021)Recognition and Analysis of Image Patterns Based on Latin Squares by Means of Computational Algebraic GeometryMathematics10.3390/math90606669:6(666)Online publication date: 21-Mar-2021
Computing Autotopism Groups of Partial Latin Rectangles
Article No.: 1.12, Pages 1 - 39
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.
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Index Terms
- Computing Autotopism Groups of Partial Latin Rectangles
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Publication History
Published: 30 September 2020
Accepted: 01 July 2020
Revised: 01 May 2020
Received: 01 October 2019
Published in JEA Volume 25
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- NSF of China
- NSFC Research Fellowship for International Young Scientists
- Science and Technology Development Plan of Tianjin
- Junta de Andalucía and the Departmental Research Budget of the Department of Applied Mathematics I of the University of Seville
- Fundamental Research Funds for the Central Universities and SAFEA: Overseas Young Talents in Cultural and Educational Sector
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View all- Falcón R(2021)Recognition and Analysis of Image Patterns Based on Latin Squares by Means of Computational Algebraic GeometryMathematics10.3390/math90606669:6(666)Online publication date: 21-Mar-2021
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