Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Sloane, N.: The on-line encyclopedia of integer sequences. https://oeis.org/
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 Squares
Egan, J., Wanless, I.M.: Enumeration of MOLS of small order. Math. Comput. 85(298), 799–824 (2016)
McGuire, G., Tugemann, B., Civario, G.: There is no 16-clue Sudoku: solving the Sudoku minimum number of clues problem via hitting set enumeration. Exp. Math. 23(2), 190–217 (2014)
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_9
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)
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)
Pickover, C.A.: The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions. Princeton University Press, Princeton (2002)
Hulpke, A., Kaski, P., Östergárd, P.: The number of Latin squares of order 11. Math. Comput. 80(274), 1197–1219 (2011)
McKay, B.D., Rogoyski, E.: Latin squares of order 10. Electr. J. Comb. 2(1), 1–4 (1995)
McKay, B.D., Wanless, I.M.: On the number of Latin squares. Ann. Comb. 9(3), 335–344 (2005)
Trotter, H.F.: Algorithm 115: perm. Commun. ACM 5(8), 434–435 (1962)
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)
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)
Zaikin, O., Zhuravlev, A., Kochemazov, S., Vatutin, E.: On the construction of triples of diagonal Latin squares of order 10. Electron. Notes Discrete Math. 54, 307–312 (2016)
Acknowledgements
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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Vatutin, E., Belyshev, A., Kochemazov, S., Zaikin, O., Nikitina, N. (2019). Enumeration of Isotopy Classes of Diagonal Latin Squares of Small Order Using Volunteer Computing. In: Voevodin, V., Sobolev, S. (eds) Supercomputing. RuSCDays 2018. Communications in Computer and Information Science, vol 965. Springer, Cham. https://doi.org/10.1007/978-3-030-05807-4_49
Download citation
DOI: https://doi.org/10.1007/978-3-030-05807-4_49
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05806-7
Online ISBN: 978-3-030-05807-4
eBook Packages: Computer ScienceComputer Science (R0)