The Journal of the Institute of Internet, Broadcasting and Communication (한국인터넷방송통신학회논문지)

The Institute of Internet, Broadcasting and Communication (한국인터넷방송통신학회)
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Binary Backtracking Algorithm for Sudoku

스도쿠 퍼즐을 위한 이진역추적 알고리즘

  • Lee, Sang-Un (Dept. of Multimedia Eng., Gangneung-Wonju National University)
  • 이상운 (강릉원주대학교 과학기술대학 멀티미디어공학과)
  • Received : 2017.05.28
  • Accepted : 2017.08.11
  • Published : 2017.08.31
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Abstract

This paper suggests polynomial time solution algorithm for Sudoku puzzle problem. This problem has been known NP (non-deterministic polynomial time)-complete. The proposed algorithm set the initial value of blank cells to value range of [$1,2,{\cdots},9$]. Then the candidate set values in blank cells deleted by preassigned clue in row, column, and block. We apply the basic rules of Stuart, and proposes two additional rules. Finally we apply binary backtracking(BBT) technique. For the experimental Sudoku puzzle with various categories of solution, the BBT algorithm can be obtain all of given Sudoku puzzle regardless of any types of solution.

본 논문은 지금까지 NP-완전 문제로 다항시간 알고리즘이 존재하지 않는 스도쿠 퍼즐 문제의 해를 다항시간으로 구하는 알고리즘을 제안하였다. 제안된 알고리즘은 빈칸들에 [$1,2,{\cdots},9$] 중에서 행, 열과 블록에 존재하는 실마리 숫자를 제외한 후보 집합을 초기치로 설정하였다. 빈칸의 후보 집합에 대해 Stuart이 제시한 기본적인 규칙들과 더불어 2개의 추가 규칙을 제시하고, 마지막으로 이진 역추적 기법(BBT)을 적용하였다. 다양한 부류의 해를 갖는 실험데이터들에 대해 적용한 결과 제안된 BBT 알고리즘은 어떠한 부류의 해를 갖던지에 상관없이 주어진 스도쿠 퍼즐을 풀 수 있음을 보였다.

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References

  1. G. McGuire, B. Tugemann, and G. Givario, "There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem," School of Mathematical Sciences, University College Dublin, Ireland, pp. 1-36, Jan, 2012. arXiv: 1201.0749
  2. The Physics arXiv Blog, "Mathematicians Solve Minimum Sudoku Problem," http://www.technologyreview.com/view/509091/best-of-2012-mathematicians-solve-minimum-sudoku-problem/, MIT Technology Review, Jan. 2012.
  3. J. P. Delahaye, "The Science Behind Sudoku," Scientific American, Vol. 294, No. 6, pp. 80-87 Jun. 2006, doi:10.1038/scientificamerican0606-80
  4. H. H. Lin, and I. C. Wu, "Solving the Minimum Sudoku Problem," International Conference on Technologies and Applications of Artificial Intelligence (TAAI), pp. 456-461, Nov. 2010, doi:10.1109/TAAI.2010.77
  5. A. M. Herzberg and M. R. Murty, "Sudoku Squares and Chromatic Polynomials," Notices of the American Mathematical Society (AMS), Vol. 54, No. 6, pp. 708-717, Jun. 2007.
  6. B. Felgenhauer and F. Jarvis, "Enumerating Possible Sudoku Grids," http://www.afjarvis.staff.shef.ac.uk/sudoku.pdf, 2005.
  7. T. Yato and T. Seta, "Complexity and Completeness of Finding Another Solution and Its Application to Puzzles," IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol .E86-A, No. 5, pp. 1052-1060, May 2003, ISSN: 0916-8508
  8. E. R. Maria and Z. Toroczkai, "The Chaos Within Sudoku," Scientific Reports, Vol. 2, pp. 1-8, Oct. 2012, doi:10.1038/srep00725.
  9. R. Lewis, "Metaheuristics Can Solve Sudoku Puzzles," Journal of Heuristics, Vol. 13, No. 4, pp. 387-401, Aug. 2007, doi:10.1007/s10732-007-9012-8
  10. E. S. Reich, "Mathematician Clams Breakthrough in Sudoku Puzzle," Nature News, Jan. 2012, doi:10.10.1038/nature.2012.9751
  11. Wikipedia, "Algorithmics of Sudoku," http://en.wikipedia.org/wiki/Algorithmics_of_Sudoku, Wikipedia Foundation Ltd., 2015.
  12. J. F. Crook, "A Pencil-and-Paper Algorithm for Solving Sudoku Puzzles," Notices of the American Mathematical Society (AMS), Vol. 56, No. 4, pp. 460-468, Apr. 2009.
  13. A. Stuart, "Sudoku Solver for Android and iPhone: Sudoku Solver for Mobile Devices. Ver. 1.97," http://www.sudokuwi ki.or g/sudoku.htm, Syndicated Puzzles Inc, Jul. 2013.
  14. Sudoku-Solutions.com, "Sudoku Solutions: Sudoku Solver," http://www.sudoku-solutions.com/, Aire Technologies, Wiesbaden, Germany, 2013.
  15. S. U. Lee, "Sudoku Algorithm," Journal of the IIBC, Vol. 15, No. 1, pp. 207-215, Feb. 2015, doi: 10.7236/JIIBC.2015.15.1.207
  16. M. Swain, "World's Hardest Sudoku Puzzle: It's the Most Baffling Brainteaser Ever Devised... Can You Solve it?," http://www.mirror.co.uk/news/weired-news/worlds-hardest-sudoku-puzzle-ever-942299, Jun. 2012.