Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Computing Random r-Orthogonal Latin Squares

  • Conference paper
  • First Online:
Combinatorial Optimization and Applications (COCOA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14462))

  • 371 Accesses

Abstract

Two Latin squares of order n are r-orthogonal if, when superimposed, there are exactly r distinct ordered pairs. The spectrum of all values of r for Latin squares of order n is known. A Latin square A of order n is r-self-orthogonal if A and its transpose are r-orthogonal. The spectrum of all values of r is known for all orders \(n\ne 14\). We develop randomized algorithms for computing pairs of r-orthogonal Latin squares of order n and algorithms for computing r-self-orthogonal Latin squares of order n.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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 79.99
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

Similar content being viewed by others

Notes

  1. 1.

    It is based on the Java implementation described by Ignacio Gallego Sagastume

    https://github.com/bluemontag/igs-lsgp.

References

  1. Arce-Nazario, R.A., Castro, F.N., Córdova, J., Hicks, K., Mullen, G.L., Rubio, I.M.: Some computational results concerning the spectrum of sets of Latin squares. Quasigroups Related Syst. 22, 159–164 (2014)

    MathSciNet  Google Scholar 

  2. Asratyan, A., Mirumyan, A.: Transformations of Latin squares (Russian). Diskret. Mat. 2, 21–28 (1990)

    MathSciNet  Google Scholar 

  3. Belyavskaya, G.B.: \(r\)-orthogonal quasigroups I. Math. Issled. 39, 32–39 (1976)

    MathSciNet  Google Scholar 

  4. Belyavskaya, G.B.: \(r\)-orthogonal quasigroups II. Math. Issled. 43, 39–49 (1977)

    MathSciNet  Google Scholar 

  5. Belyavskaya, G.B.: \(r\)-orthogonal Latin squares. In: DL enes, A.K.J., editor, Latin Squares: New Developments, pp. 169–202 (Chapter 6). Elsevier (1992)

    Google Scholar 

  6. Burkard, R.E., Çela, E.: Linear assignment problems and extensions. In: Du, D., Pardalos, P.M., editors, Handbook of Combinatorial Optimization, pp. 75–149. Springer (1999). https://doi.org/10.1007/978-1-4757-3023-4_2

  7. Colbourn, C.J.: The complexity of completing partial Latin squares. Discret. Appl. Math. 8(1), 25–30 (1984)

    Article  MathSciNet  Google Scholar 

  8. Colbourn, C.J., Zhu, L.: The spectrum of \(r\)-orthogonal Latin squares. In: Colbourn, C.J., Mahmoodian, E.S. (eds.) Combinatorics Adv., pp. 49–75. Springer, US, Boston, MA (1995)

    Chapter  Google Scholar 

  9. Dinitz, J.H., Stinson, D.R.: On the maximum number of different ordered pairs of symbols in sets of Latin squares. J. Comb. Des. 13(1), 1–15 (2005)

    Article  MathSciNet  Google Scholar 

  10. Egan, J., Wanless, I.M.: Latin squares with restricted transversals. J. Comb. Des. 20(7), 344–361 (2012)

    Article  MathSciNet  Google Scholar 

  11. Euler, L.: Recherche sur une nouvelle espéce de quarrés magiques. Leonardi Euleri Opera Omnia 7, 291–392 (1923)

    Google Scholar 

  12. Evans, A.B.: Latin squares without orthogonal mates. Des. Codes Cryptogr. 40(1), 121–130 (2006)

    Article  MathSciNet  Google Scholar 

  13. Finizio, N.J., Zhu, L.: Self-orthogonal Latin squares (SOLS). In: Colbourn, C.J., Dinitz, J.H., editors, The Handbook of Combinatorial Designs, pp. 211–219. Chapman/CRC Press (2007)

    Google Scholar 

  14. Hall, M.: An existence theorem for Latin squares. Bull. Amer. Math. Soc. 51(6), 387–388 (1945)

    Article  MathSciNet  Google Scholar 

  15. Hall, P.: On representative of subsets. J. London Math. Soc. 10, 26–30 (1935)

    Article  MathSciNet  Google Scholar 

  16. Hankin, P.: Generating random Latin squares (blog). https://blog.paulhankin.net/latinsquares/ (2019)

  17. Jacobson, M.T., Matthews, P.: Generating uniformly distributed random Latin squares. J. Comb. Des. 4(6), 405–437 (1996)

    Article  MathSciNet  Google Scholar 

  18. Keedwell, A.D., Dénes, J.: Latin squares and their applications. Elsevier (2015)

    Google Scholar 

  19. Keedwell, A.D., Mullen, G.L.: Sets of partially orthogonal Latin squares and projective planes. Discret. Math. 288(1–3), 49–60 (2004)

    Article  MathSciNet  Google Scholar 

  20. Kuhn, H.W.: The Hungarian method for the assignment problem. Naval Res. Logist. Quart. 2(1–2), 83–97 (1955)

    Article  MathSciNet  Google Scholar 

  21. Mann, H.B.: On orthogonal Latin squares. Bull. Amer. Math. Soc. 50, 249–257 (1944)

    Article  MathSciNet  Google Scholar 

  22. Mariot, L., Formenti, E., Leporati, A.: Constructing orthogonal Latin squares from linear cellular automata. CoRR, abs/1610.00139 (2016)

    Google Scholar 

  23. Mariot, L., Gadouleau, M., Formenti, E., Leporati, A.: Mutually orthogonal Latin squares based on cellular automata. Des. Codes Cryptogr. 88(2), 391–411 (2020)

    Article  MathSciNet  Google Scholar 

  24. McKay, B.D., Meynert, A., Myrvold, W.: Small Latin squares, quasigroups, and loops. J. Comb. Des. 15(2), 98–119 (2007)

    Article  MathSciNet  Google Scholar 

  25. van Rees, G.H.J.: Subsquares and transversals in Latin squares. Ars Combin. 29B, 193–204 (1990)

    MathSciNet  Google Scholar 

  26. Wanless, I.M.: Cycle switches in Latin squares. Graphs Comb. 20(4), 545–570 (2004)

    Article  MathSciNet  Google Scholar 

  27. Wanless, I.M., Webb, B.S.: The existence of Latin squares without orthogonal mates. Des. Codes Cryptogr. 40(1), 131–135 (2006)

    Article  MathSciNet  Google Scholar 

  28. Zhang, H.: 25 new r-self-orthogonal Latin squares. Discret. Math. 313(17), 1746–1753 (2013)

    Article  MathSciNet  Google Scholar 

  29. Zhu, L., Zhang, H.: Completing the spectrum of r-orthogonal Latin squares. Discret. Math. 268(1–3), 343–349 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Erik Jonsson School of Engineering and Computer Science, University of Texas at Dallas, Richardson, USA

    Sergey Bereg

Authors
  1. Sergey Bereg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey Bereg .

Editor information

Editors and Affiliations

  1. University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

  2. Beijing Normal University, Zhuhai, China

    Jianxiong Guo

Rights and permissions

Reprints and permissions

Copyright information

© 2024 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

Bereg, S. (2024). Computing Random r-Orthogonal Latin Squares. In: Wu, W., Guo, J. (eds) Combinatorial Optimization and Applications. COCOA 2023. Lecture Notes in Computer Science, vol 14462. Springer, Cham. https://doi.org/10.1007/978-3-031-49614-1_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-49614-1_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-49613-4

  • Online ISBN: 978-3-031-49614-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics