Advertisement

Journal of Global Optimization

, Volume 51, Issue 3, pp 497–514 | Cite as

New decision rules for exact search in N-Queens

  • Pablo San Segundo
Article

Abstract

This paper presents a set of new decision rules for exact search in N-Queens. Apart from new tiebreaking strategies for value and variable ordering, we introduce the notion of ‘free diagonal’ for decision taking at each step of the search. With the proposed new decision heuristic the number of subproblems needed to enumerate the first K solutions (typically K = 1, 10 and 100) is greatly reduced w.r.t. other algorithms and constitutes empirical evidence that the average solution density (or its inverse, the number of subproblems per solution) remains constant independent of N. Specifically finding a valid configuration was backtrack free in 994 cases out of 1,000, an almost perfect decision ratio. This research is part of a bigger project which aims at deriving new decision rules for CSP domains by evaluating, at each step, a constraint value graph G c . N-Queens has adapted well to this strategy: domain independent rules are inferred directly from G c whereas domain dependent knowledge is represented by an induced hypergraph over G c and computed by similar domain independent techniques. Prior work on the Number Place problem also yielded similar encouraging results.

Keywords

Decision heuristics N-Queens Global search Constraint satisfaction Maximum clique 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramson B., Yung M.: Divide and conquer under global constraints: a solution to the N-queens problem. J. Parallel Distrib. Comput. 6(3), 649–660 (1989)CrossRefGoogle Scholar
  2. 2.
    Bell J., Stevens B.: A survey of known results and research areas for N-Queens. Discret. Math. 309(1), 1–31 (2009) ElsevierCrossRefGoogle Scholar
  3. 3.
    Berge, C.: Graphes et hypergraphe. Monographies Universitaires de Mathematiques (37), Dunod, Paris (1970)Google Scholar
  4. 4.
    Bernhardsson, B.: Explicit solutions to the N-Queens problems for all n. ACM SIGART Bull. 2(7); ACM Press (1991)Google Scholar
  5. 5.
    Bitner J.R., Reingold E.M.: Backtrack programming techniques. Commun. ACM 18(11), 651–656 (1975)CrossRefGoogle Scholar
  6. 6.
    Chen, J.: An efficient non-probabilistic search algorithm for the N-Queens problem. In: Proceedings on the Third International Conference: Advances in Computer Science and Technologies (IASTED 07), pp. 393–396, Thailand (2007)Google Scholar
  7. 7.
    A leading answer set problem solver. URL: http://www.cs.uni-potsdam.de/clasp/Download/clasp-1.1.0.tar.gz
  8. 8.
    Crawford, K.D.: Solving the N-Queens problem using genetic algorithms. In: Berghel, H., Hedrick, G., Deaton, E., Roach, D., Wainwright, R. (eds.), Proceedings of the 1992 ACM/SIGAPP Symposium on Applied Computing: Technological Challenges of the 1990’s. SAC ‘92, pp. 1039–1047. ACM (1992)Google Scholar
  9. 9.
    Dechter R., Pearl J.: A problem simplification approach that generates heuristics for constraint-satisfaction problems. In: Hayes, J.E., Michie, D., Richards, J. (eds) Machine Intelligence, 11, pp. 125–155. Oxford University Press, New York (1988)Google Scholar
  10. 10.
  11. 11.
    Du, D., Pardalos, P. (eds): Hand Book of Combinatorial Optimization, Supplement Volume A, pp. 421–424. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  12. 12.
    Erbas C., Tanik M., Nair V.S.S.: Parallel memory allocation and data alignment in SIMD machines. Parallel Algorithms and Applications 4(1), 139–151 (1994)Google Scholar
  13. 13.
    Erbas, C., Sarkeshik, S., Tanik, M.M.: Different perspectives of the n-queens problem. In: Proceedings of the 1992 ACM Annual Conference on Communications, ACM Press, pp. 99–108 (1992)Google Scholar
  14. 14.
    Haralick R.M., Elliot G.L.: Increasing tree search efficiency for constraint satisfaction problems. Artif. Intell. 14, 263–314 (1980)CrossRefGoogle Scholar
  15. 15.
    Hsiang, J., Shieh, Y., Chen, Y.: The cyclic complete mapppings counting problems. Federated Logic Conference (2002, July 20–August 1), CopenhagenGoogle Scholar
  16. 16.
    Jagota, A.: Optimization by Reduction to Maximum Clique. In: Proceedings of Int’l Conference Neural Networks, pp. 1526–1531 (1993)Google Scholar
  17. 17.
    Johnston, M.D.: Scheduling with neural networks—the case of the Hubble Space Telescope. NASA Memo (1989)Google Scholar
  18. 18.
    Kale L.V.: An almost perfect heuristic for the N nonattacking queens problem. Inf. Process. Lett. 34(1), 73–178 (1990)Google Scholar
  19. 19.
    Karp, R.M.: In:Miller, R.E., Thatcher, J.W. (eds.) Reducibility among Combinatorial Problems, pp. 85–103. Plenum, New York (1972)Google Scholar
  20. 20.
    Katsirelos, G., Bacchus, F.: Generalized nogoods in CSPs. In: Proceedings of the 20th National Conference on Artificial Intelligence, vol. 1. AAAI Press, pp. 390–396 (2005)Google Scholar
  21. 21.
    Knuth D.: Dancing links. In: Davies, J., Roscoe, B., Woodcock, J. (eds) Millenial Perspectives in Computer Science, pp. 187–214. Palgrave, Hampshire (2000)Google Scholar
  22. 22.
    Kunde, M., Gürtzig, K.: Efficient sorting and routing on reconfigurable meshes using restricted bus length. In: Proceedings of the 11th International Parallel Processing Symposium (IPPS 1997), Switzerland. IEEE Computer Society, pp. 713–720 (1997)Google Scholar
  23. 23.
    Gent, I.P., Jefferson, C., Miguel, I.: Minion: a fast scalable constraint solver. In: Proceedings of ECAI 2006. IOS Press, pp. 98–102 (2006)Google Scholar
  24. 24.
    Minton, S., Johnston, M.D., Philips, A.B., Laird, P.: Solving large-scale constraint-satisfaction and scheduling problems using a heuristic repair method. In: Proceedings National Conference on Artificial Intelligence (AAAI), pp. 17–24 (1990)Google Scholar
  25. 25.
    Purdom P.W.: Search rearrangement backtracking and polynomial average time. Artif. Intell. 21(1–2), 117–133 (1983)CrossRefGoogle Scholar
  26. 26.
    San Segundo, P., Rodriguez-Losada, D., Galán, R., Matía, F., Jiménez, A.: Exploiting CPU bit parallel operations to improve efficiency in search. In: Proceedings of International Conference on Tools for Atificial Intelligence (ICTAI 07). Patrás, Greece (October 2007)Google Scholar
  27. 27.
    San Segundo, P., Jiménez, A.: Using graphs to derive CSP heuristics and its application to Sudoku. In: Proceedings of International Conference on Tools for Atificial Intelligence (ICTAI 09), New York, USA (November 2009)Google Scholar
  28. 28.
    Simonis, H.: Chapter 6 of ECLIPSe ELearning course, Cork Constraint Computation Centre, url:http://4c.ucc.ie/~hsimonis/ELearning/index.htm
  29. 29.
    Sosic, R., Gu, J.: Efficient local search with conflict minimization. IEEE Trans, on Knowledge and Data Engineering (6E), 661–668 (Oct. 1994)Google Scholar
  30. 30.
    Sosic R., Gu J.: 3,000,000 queens in less than a minute. ACM SIGART Bulletin 2(2), 22–24 (1991) ACM PressCrossRefGoogle Scholar
  31. 31.
    Stone H.S., Sipala P.: The average complexity of depth-first search with backtracking and cutoff. IBM J. Res. Dev. 30(3), 242–258 (1986)CrossRefGoogle Scholar
  32. 32.
    Stone, H.S., Sone, J.M.: Efficient search techniques—an empirical study of the N-Queens problem. IBM Journal of Research and Development 31(4), 464–474 1987. ISSN: 0018-8646Google Scholar
  33. 33.
    Tanik, M.M.: A Graph Model for Deadlock Prevention. Doctoral Thesis. UMI Order Number: AAI7909241. Texas A&M University (1978)Google Scholar
  34. 34.
    Yamamoto T., Jinno K., Hirose H.: A dynamical N queen problem solver using hysteresis neural networks. IEICE Transactions on Fundamentals of Electronics Communications and Computer Science 86(4), 740–745 (2003)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Centro de Automática y Robótica (CAR)Universidad Politécnica de Madrid, UPMMadridSpain

Personalised recommendations