Skip to main content
SpringerLink
Log in

Classification of Cells Mapping Schemes Related to Orthogonal Diagonal Latin Squares of Small Order

  • Conference paper
  • First Online:
Supercomputing (RuSCDays 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14389))

Included in the following conference series:

  • 144 Accesses

Abstract

Volunteer computing is a cheap yet efficient type of distributed computing, where desktops of private persons are united into projects. Some of these projects are aimed at finding new mathematical objects based on orthogonal systems of Latin squares. In 2021, new systems of orthogonal diagonal Latin squares of order 10 were found in a volunteer computing project. This was done using cells mapping schemes related to extended self-orthogonal diagonal Latin squares. In the present study, a classification of such schemes is proposed. The classification is built upon a structure of a multiset of cycle lengths, when a scheme is considered a permutation. For orders 1–9, the classification is constructed completely on a computer, while for order 10 only some classes are determined via volunteer computing. Finally, for order 10 new orthogonal systems were investigated using the cells mapping schemes and a SAT solver on a computer. It is described how the latter results can lead to finding the remaining classes for order 10 via volunteer computing.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 74.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    https://oeis.org/A309210.

References

  1. Anderson, D.P.: BOINC: a platform for volunteer computing. J. Grid Comput. 18(1), 99–122 (2020)

    Article  Google Scholar 

  2. Biere, A., Fleury, M.: Gimsatul, IsaSAT and Kissat entering the SAT Competition 2022. In: Proceedings of SAT Competition 2022 - Solver and Benchmark Descriptions, pp. 10–11 (2022)

    Google Scholar 

  3. Knuth, D.E.: Dancing Links. Millenial Perspectives in Computer Science, pp. 187–214 (2000)

    Google Scholar 

  4. Kochemazov, S., Zaikin, O., Semenov, A.: The comparison of different SAT encodings for the problem of search for systems of orthogonal Latin squares. In: Proceedings of the International Conference on Mathematical and Information Technologies, pp. 155–165 (2017)

    Google Scholar 

  5. Kochemazov, S., Zaikin, O., Vatutin, E., Belyshev, A.: Enumerating diagonal Latin squares of order up to 9. J. Integer Seq. 23(1), 20.1.2 (2020)

    Google Scholar 

  6. Laywine, C., Mullen, G.: Discrete Mathematics Using Latin Squares. Wiley-Interscience, New York (1998)

    Google Scholar 

  7. Marques-Silva, J.P., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999)

    Article  MathSciNet  Google Scholar 

  8. Parker, E.T.: Orthogonal Latin squares. Proc. Natl. Acad. Sci. U.S.A. 45(6), 859–862 (1959)

    Article  MathSciNet  Google Scholar 

  9. Sloane, N.J.A.: The on-line encyclopedia of integer sequences. Electron. J. Comb. 1 (1994)

    Google Scholar 

  10. Vatutin, E.I., Belyshev, A.D., Nikitina, N.N., Manzuk, M.O.: Usage of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares. Telecommunications 1(1), 2–16 (2023)

    Article  Google Scholar 

  11. Vatutin, E., Belyshev, A.: Enumerating the orthogonal diagonal Latin squares of small order for different types of orthogonality. In: Proceedings of the 6th Russian Supercomputing Days, pp. 586–597 (2020)

    Google Scholar 

  12. Vatutin, E., Belyshev, A., Kochemazov, S., Zaikin, O., Nikitina, N.: Enumeration of isotopy classes of diagonal Latin squares of small order using volunteer computing. In: Proceedings of the 4th Russian Supercomputing Days, pp. 578–586 (2018)

    Google Scholar 

  13. Vatutin, E., Titov, V., Zaikin, O., Kochemazov, S., Manzuk, M., Nikitina, N.: Orthogonality-based classification of diagonal Latin squares of order 10. In: Proceedings of the VIII International Conference on Distributed Computing and Grid-Technologies in Science and Education, pp. 282–287 (2018)

    Google Scholar 

  14. 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)

    Article  Google Scholar 

  15. Vatutin, E., Zaikin, O., Manzyuk, M., Nikitina, N.: Searching for orthogonal Latin squares via cells mapping and BOINC-based Cube-and-Conquer. In: Proceedings of the 7th Russian Supercomputing Days, pp. 498–512 (2021)

    Google Scholar 

  16. Zhang, H.: Combinatorial designs by SAT solvers. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability - Second Edition, Frontiers in Artificial Intelligence and Applications, vol. 336, pp. 819–858. IOS Press (2021)

    Google Scholar 

Download references

Acknowledgements

Authors are grateful to volunteers who provided their computational resources to Gerasim@home. Oleg Zaikin was funded by the Ministry of Education of the Russian Federation, project No. 121041300065-9.

Author information

Authors and Affiliations

  1. Southwest State University, Kursk, Russia

    Eduard Vatutin

  2. ISDCT SB RAS, Irkutsk, Russia

    Oleg Zaikin

Authors
  1. Eduard Vatutin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oleg Zaikin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oleg Zaikin .

Editor information

Editors and Affiliations

  1. Research Computing Center (RCC), Moscow State University, Moscow, Russia

    Vladimir Voevodin

  2. Research Computing Center (RCC), Moscow State University, Moscow, Russia

    Sergey Sobolev

  3. Russian Academy of Sciences (RAS), Keldysh Institute of Applied Mathematics, Moscow, Russia

    Mikhail Yakobovskiy

  4. Russian Federal Nuclear Center, Sarov, Russia

    Rashit Shagaliev

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Vatutin, E., Zaikin, O. (2023). Classification of Cells Mapping Schemes Related to Orthogonal Diagonal Latin Squares of Small Order. In: Voevodin, V., Sobolev, S., Yakobovskiy, M., Shagaliev, R. (eds) Supercomputing. RuSCDays 2023. Lecture Notes in Computer Science, vol 14389. Springer, Cham. https://doi.org/10.1007/978-3-031-49435-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-49435-2_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-49434-5

  • Online ISBN: 978-3-031-49435-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation