Abstract

Harmony search (HS) algorithm was applied to solving Sudoku puzzle. The HS is an evolutionary algorithm which mimics musicians’ behaviors such as random play, memory-based play, and pitch-adjusted play when they perform improvisation. Sudoku puzzles in this study were formulated as an optimization problem with number-uniqueness penalties. HS could successfully solve the optimization problem after 285 function evaluations, taking 9 seconds. Also, sensitivity analysis of HS parameters was performed to obtain a better idea of algorithm parameter values.

Keywords

Sudoku puzzle harmony search combinatorial optimization 

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References

  1. 1.
    Eppstein, D.: Nonrepetitive Paths and Cycles in Graphs with Application to Sudoku. ACM Computing Research Repository. cs.DS/0507053  (2005)Google Scholar
  2. 2.
    Caine, A., Cohen, R.: A Mixed-Initiative Intelligent Tutoring System for Sudoku. In: Lamontagne, L., Marchand, M. (eds.) Canadian AI 2006. LNCS (LNAI), vol. 4013, pp. 550–561. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Nicolau, M., Ryan, C.: Solving Sudoku with the GAuGE System. In: Collet, P., Tomassini, M., Ebner, M., Gustafson, S., Ekárt, A. (eds.) EuroGP 2006. LNCS, vol. 3905, pp. 213–224. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Yato, T., Seta, T.: Complexity and Completeness of Finding Another Solution and its Application to Puzzles. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences 86, 1052–1060 (2003)Google Scholar
  5. 5.
    Geem, Z.W., Kim, J.H., Loganathan, G.V.: A New Heuristic Optimization Algorithm: Harmony Search. Simulation 76(2), 60–68 (2001)Google Scholar
  6. 6.
    Lee, K.S., Geem, Z.W.: A New Structural Optimization Method Based on the Harmony Search Algorithm. Computers and Structures 82(9-10), 781–798 (2004)CrossRefGoogle Scholar
  7. 7.
    Geem, Z.W.: Optimal Cost Design of Water Distribution Networks using Harmony Search. Engineering Optimization 38(3), 259–280 (2006)CrossRefGoogle Scholar
  8. 8.
    Geem, Z.W.: Optimal Scheduling of Multiple Dam System Using Harmony Search Algorithm. In: Lecture Notes in Computer Science, vol. 4507, pp. 316–323 (2007)Google Scholar
  9. 9.
    Geem, Z.W., Lee, K.S., Park, Y.: Application of Harmony Search to Vehicle Routing. American Journal of Applied Sciences 2(12), 1552–1557 (2005)CrossRefGoogle Scholar
  10. 10.
    Geem, Z.W., Hwangbo, H.: Application of Harmony Search to Multi-Objective Optimization for Satellite Heat Pipe Design. In: Proceedings of 2006 US-Korea Conference on Science, Technology, & Entrepreneurship (UKC 2006). CD-ROM (2006)Google Scholar
  11. 11.
    Ryu, S., Duggal, A.S., Heyl, C.N., Geem, Z.W.: Mooring Cost Optimization via Harmony Search. In: Proceedings of the 26th International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2007), ASME. CD-ROM (2007)Google Scholar
  12. 12.
    Kim, J.H., Geem, Z.W., Kim, E.S.: Parameter Estimation of the Nonlinear Muskingum Model Using Harmony Search. Journal of the American Water Resources Association 37(5), 1131–1138 (2001)CrossRefGoogle Scholar
  13. 13.
    Geem, Z.W., Choi, J.–Y.: Music Composition Using Harmony Search Algorithm. In: Lecture Notes in Computer Science, vol. 4448, pp. 593–600 (2007)Google Scholar
  14. 14.
    Web Sudoku (January 19, 2007), http://www.websudoku.com/

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Zong Woo Geem
    • 1
  1. 1.Johns Hopkins University, Environmental Planning and Management Program, 729 Fallsgrove Drive #6133, Rockville, Maryland 20850USA

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