Enumeration of Isotopy Classes of Diagonal Latin Squares of Small Order Using Volunteer Computing
The paper is devoted to discovering new features of diagonal Latin squares of small order. We present an algorithm, based on a special kind of transformations, that constructs a canonical form of a given diagonal Latin square. Each canonical form corresponds to one isotopy class of diagonal Latin squares. The algorithm was implemented and used to enumerate the isotopy classes of diagonal Latin squares of order at most 8. For order 8 the computational experiment was conducted in a volunteer computing project. The algorithm was also used to estimate how long it would take to enumerate the isotopy classes of diagonal Latin squares of order 9 in the same volunteer computing project.
KeywordsVolunteer computing Combinatorics Latin square Diagonal Latin square Enumeration
The research was partially supported by Russian Foundation for Basic Research (grants 16-07-00155-a, 17-07-00317-a, 18-07-00628-a, 18-37-00094-mol-a) and by Council for Grants of the President of the Russian Federation (stipend SP-1829.2016.5).
- 1.Sloane, N.: The on-line encyclopedia of integer sequences. https://oeis.org/
- 2.Colbourn, C., et al.: Handbook of Combinatorial Designs. Discrete Mathematics and Its Applications, 2nd edn, pp. 224–265. Chapman and Hall/CRC, London (2006). chap. Latin SquaresGoogle Scholar
- 5.Vatutin, E.I., Kochemazov, S.E., Zaikin, O.S.: Applying volunteer and parallel computing for enumerating diagonal Latin squares of order 9. In: Sokolinsky, L., Zymbler, M. (eds.) PCT 2017. CCIS, vol. 753, pp. 114–129. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-67035-5_9CrossRefGoogle Scholar
- 6.Vatutin, E., Zaikin, O., Zhuravlev, A., Manzyuk, M., Kochemazov, S., Titov, V.S.: Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares. In: CEUR Workshop Proceedings, Selected papers of the 7th International Conference on Distributed Computing and Grid-Technologies in Science and Education, vol. 1787, pp. 486–490 (2017)Google Scholar
- 7.Vatutin, E., Zaikin, O., Kochemazov, S., Valyaev, S.: Using volunteer computing to study some features of diagonal Latin squares. Open Eng. 7, 453–460 (2017)Google Scholar
- 13.Vatutin, E., Valyaev, S., Titov, V.: Comparison of sequential methods for getting separations of parallel logic control algorithms using volunteer computing. In: Second International Conference BOINC-based High Performance Computing: Fundamental Research and Development (BOINC:FAST 2015), Petrozavodsk, Russia, September 14–18, 2015, vol. 1502, pp. 37–51. CEUR-WS (2015)Google Scholar
- 14.Brown, J., Cherry, F., Most, L., Parker, E., Wallis, W.: Completion of the spectrum of orthogonal diagonal Latin squares. Lecture Notes in Pure and Applied Mathematics, vol. 139, pp. 43–49 (1992)Google Scholar