Abstract
The article describes a number of simple transformations (rotation of intercalates, loops, Latin subrectangles, replacement of transversals) in the problem of constructing a collection of orthogonal diagonal Latin squares (ODLS) of order 10 aiming to try to find a triple of pairwise orthogonal diagonal Latin squares of order 10. The averaged time spent on obtaining one canonical form of ODLS using efficient software implementations of square generators based on nested loops and bit arithmetic, on the one hand, and the Euler-Parker method for checking the DLS for the presence of ODLS (together with the composition of the canonizer, if necessary), on the other hand, is 8.3 h using single threaded CPU implementation. It is shown that the application of the indicated transformations in some cases is capable of providing new canonical forms (CFs) of ODLS with significantly lower computational time. The article presents the results of comparing the effectiveness of simple transformations, as well as estimates of the minimum and maximum number of certain structural elements of the DLS (intercalates, loops, Latin subrectangles) depending on its order N (sequences A307163, A307164, A307166, A307167, A307170, A307171, A307841, A307842, A307839, A307840, A287645, A287644 in OEIS). The postprocessor of the found CF of ODLS, working within the framework of the Gerasim@Home volunteer distributed computing project based on this subset of effective simple transformations, allows increasing the output of new CF of ODLS by 10–15% and the required computational costs – by 2–3%.
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Acknowledgments
The article partially supported by RFBR, grant number 18-07-00628. The authors of the article would like to thank all the crunchers who took part in the Gerasim@Home volunteer distributed computing project, as well as citerra and SerVal members (the Russia Team) of the BOINC.ru Internet portal for their help in organizing computational experiments and constructive discussion of their details.
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Vatutin, E., Belyshev, A., Nikitina, N., Manzuk, M. (2020). Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10. In: Jordan, V., Filimonov, N., Tarasov, I., Faerman, V. (eds) High-Performance Computing Systems and Technologies in Scientific Research, Automation of Control and Production. HPCST 2020. Communications in Computer and Information Science, vol 1304. Springer, Cham. https://doi.org/10.1007/978-3-030-66895-2_9
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