The Largest Critical Sets of Latin Squares

Hermiston, Keith

doi:10.1109/ciss.2019.8692907

Cited by 1 publication

(1 citation statement)

References 12 publications

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“…There are numerous other studies on counting algebraic structures which have reported incorrect counts [13], [17], [18], [20]. These historical results were obtained through deduced mathematical formulas, applying algorithmic approaches or formulating them as constraint programming problems [7], [8], [11], [24], [25]. In this paper we use constraint programming and isomorphism checking techniques to enumerate algebraic structures.…”

Section: Introductionmentioning

confidence: 99%

Efficient Enumeration of Higher Order Algebraic Structures

Khan

2020

IEEE Access

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Algebraic structures are widely studied mathematical structures in abstract algebra. Enumerating higher order algebraic structures is a computationally intensive task due to large number of possible permutations and the presence of many symmetrically equivalent redundant structures. This paper describes a comprehensive methodology and a novel algorithm to efficiently enumerate higher order algebraic structures. Enumeration of these structures is performed in two steps, namely: generation of complete set of algebraic structures with given constraints, and then computationally demanding task of performing isomorphism checking to identify isomorphism classes. In this paper we choose to study inverse property loops (IP loops), a particular class of algebraic structures, but the methodology can be applied to enumerate any algebraic structure with given constraints. IP loop constraints are modeled in Google's or-tools constraint solver to generate the complete set of IP loops of given order. The paper then discusses and evaluates several techniques to efficiently identify isomorphism classes within these algebraic structures. A novel algorithm is proposed that utilizes valid mappings and a tree data structure for efficient isomorphism checking. The algebraic structures are also modeled as color graphs and isomorphism classes are determined using state of the art graph isomorphism checking tool (nauty). The performance of these proposed isomorphism checking approaches is then evaluated using a diverse set of algebraic structure problems. It also presents an efficient use of multi-core systems to further enhance the efficiency of isomorphism checking. The proposed approach was then used to solve the previously computationally unsolved problem of determining isomorphism classes of exponent 3 IP loops of order 15. INDEX TERMS Algebraic structures, constraint solver, IP loops, isomorphism checking, isomorphism classes, nauty.

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Section: Introductionmentioning

confidence: 99%

Efficient Enumeration of Higher Order Algebraic Structures

Khan

2020

IEEE Access

View full text Add to dashboard Cite

show abstract

The Largest Critical Sets of Latin Squares

Cited by 1 publication

References 12 publications

Efficient Enumeration of Higher Order Algebraic Structures

Efficient Enumeration of Higher Order Algebraic Structures

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