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Applying Volunteer and Parallel Computing for Enumerating Diagonal Latin Squares of Order 9

  • Eduard I. Vatutin
  • Stepan E. Kochemazov
  • Oleg S. Zaikin
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 753)

Abstract

In this paper we design the algorithm aimed at fast enumeration of diagonal Latin squares of small order. This brute force based algorithm uses straightforward representation of a Latin square and greatly depends on the order in which we fill in Latin square cells. Moreover, we greatly improved its effectiveness by careful optimization using bit arithmetic. By applying this algorithm we have enumerated diagonal Latin squares of order 9. This problem was previously unsolved. In order to ensure the accuracy of the obtained result, two separate large-scale experiments were carried out. In the first one a computing cluster was employed. The second one was performed in a BOINC-based volunteer computing project. Each experiment took about 3 months. As a result we obtained two similar numbers.

Keywords

Combinatorics Latin square Enumeration Bit arithmetic Volunteer computing BOINC Parallel computing 

References

  1. 1.
    Colbourn, C.J., Dinitz, J.H.: Handbook of Combinatorial Designs, 2nd edn. (Discrete Mathematics and Its Applications). Chapman & Hall/CRC, Boca Raton (2006). http://dx.doi.org/10.1201/9781420010541 Google Scholar
  2. 2.
    Jacobson, M.T., Matthews, P.: Generating uniformly distributed random Latin squares. J. Comb. Des. 4(6), 405–437 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series, pp. 85–103. Springer, Boston (1972). doi: 10.1007/978-1-4684-2001-2_9 CrossRefGoogle Scholar
  4. 4.
    Knuth, D.: Dancing links. Millenn. Perspect. Comput. Sci. 187–214 (2000)Google Scholar
  5. 5.
    Golomb, S.W., Baumert, L.D.: Backtrack programming. J. ACM 12(4), 516–524 (1965). http://dx.doi.org/10.1145/321296.321300 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Vatutin, E., Zaikin, O., Zhuravlev, A., Manzyuk, M., Kochemazov, S., Titov, V.: Using grid systems for enumerating combinatorial objects on example of diagonal Latin squares. In: Selected Papers of the 7th International Conference Distributed Computing and Grid-technologies in Science and Education, Dubna, Russia, 4–9 July 2016. vol. 1787, pp. 486–490. CEUR-WS (2016)Google Scholar
  7. 7.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences. In: Kauers, M., Kerber, M., Miner, R., Windsteiger, W. (eds.) Calculemus/MKM -2007. LNCS, vol. 4573, p. 130. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-73086-6_12 CrossRefGoogle Scholar
  8. 8.
    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, 14–18 September 2015, vol. 1502, pp. 37–51. CEUR-WS (2015)Google Scholar
  9. 9.
    Foster, I.: Designing and Building Parallel Programs: Concepts and Tools for Parallel Software Engineering. Addison-Wesley Longman Publishing Co., Inc., Boston (1995)MATHGoogle Scholar
  10. 10.
    Anderson, D.P., Fedak, G.: The computational and storage potential of volunteer computing. In: Sixth IEEE International Symposium on Cluster Computing and the Grid (CCGrid 2006), 16–19 May 2006, Singapore, pp. 73–80. IEEE Computer Society (2006). http://dx.doi.org/10.1109/ccgrid.2006.101
  11. 11.
    Egan, J., Wanless, I.M.: Enumeration of MOLS of small order. Math. Comput. 85(298), 799–824 (2016). http://dx.doi.org/10.1090/mcom/3010 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bammel, S.E., Rothstein, J.: The number of \(9 \times 9\) Latin squares. Discrete Math. 11(1), 93–95 (1975). http://dx.doi.org/10.1016/0012-365X(75)90108-9 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    McKay, B.D., Rogoyski, E.: Latin squares of order 10. Electr. J. Comb. 2(1), 1–4 (1995)MathSciNetMATHGoogle Scholar
  14. 14.
    McKay, B.D., Wanless, I.M.: On the number of Latin squares. Ann. Comb. 9(3), 335–344 (2005). http://dx.doi.org/10.1007/s00026-005-0261-7 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Zaikin, O., Zhuravlev, A., Kochemazov, S., Vatutin, E.: On the construction of triples of diagonal Latin squares of order 10. Electron. Notes Discret. Math. 54, 307–312 (2016). http://dx.doi.org/10.1016/j.endm.2016.09.053 CrossRefMATHGoogle Scholar
  16. 16.
    Lam, C., Thiel, L., Swierz, S.: The nonexistence of finite projective planes of order 10. Canad. J. Math. 41, 1117–1123 (1989). http://dx.doi.org/10.4153/cjm-1989-049-4 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Zaikin, O., Vatutin, E., Zhuravlev, A., Manzyuk, M.: Applying high-performance computing to searching for triples of partially orthogonal Latin squares of order 10. In: 10th Annual International Scientific Conference on Parallel Computing Technologies Arkhangelsk, Russia, 29–31 March 2016, vol. 1576, pp. 155–166. CEUR-WS (2016)Google Scholar
  18. 18.
    Zaikin, O., Kochemazov, S., Semenov, A.: SAT-based search for systems of diagonal Latin squares in volunteer computing project sat@home. In: Biljanovic, P., Butkovic, Z., Skala, K., Grbac, T.G., Cicin-Sain, M., Sruk, V., Ribaric, S., Gros, S., Vrdoljak, B., Mauher, M., Tijan, E., Lukman, D. (eds.) 39th International Convention on Information and Communication Technology, Electronics and Microelectronics, MIPRO 2016, Opatija, Croatia, 30 May–3 June 2016, pp. 277–281. IEEE (2016). http://dx.doi.org/10.1109/MIPRO.2016.7522152
  19. 19.
    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). http://dx.doi.org/10.1080/10586458.2013.870056 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lin, H.H., Wu, I.C.: Solving the minimum Sudoku problem. In: The 2010 International Conference on Technologies and Applications of Artificial Intelligence, TAAI 2010, pp. 456–461 (2010). http://dx.doi.org/10.1109/taai.2010.77

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Southwest State UniversityKurskRussia
  2. 2.Matrosov Institute for System Dynamics and Control Theory SB RASIrkutskRussia

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