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Taming the Knight’s Tour: Minimizing Turns and Crossings

Authors Juan Jose Besa , Timothy Johnson, Nil Mamano , Martha C. Osegueda



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Author Details

Juan Jose Besa
  • University of California, Irvine, CA, USA
Timothy Johnson
  • University of California, Irvine, CA, USA
Nil Mamano
  • University of California, Irvine, CA, USA
Martha C. Osegueda
  • University of California, Irvine, CA, USA

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Juan Jose Besa, Timothy Johnson, Nil Mamano, and Martha C. Osegueda. Taming the Knight’s Tour: Minimizing Turns and Crossings. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 4:1-4:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.FUN.2021.4

Abstract

We introduce two new metrics of "simplicity" for knight’s tours: the number of turns and the number of crossings. We give a novel algorithm that produces tours with 9.5n+O(1) turns and 13n+O(1) crossings on a n× n board, and we show lower bounds of (6-ε)n and 4n-O(1) on the respective problems of minimizing these metrics. Hence, our algorithm achieves approximation ratios of 19/12+o(1) and 13/4+o(1). We generalize our techniques to rectangular boards, high-dimensional boards, symmetric tours, odd boards with a missing corner, and tours for (1,4)-leapers. In doing so, we show that these extensions also admit a constant approximation ratio on the minimum number of turns, and on the number of crossings in most cases.

Subject Classification

ACM Subject Classification
  • Human-centered computing → Graph drawings
  • Theory of computation → Computational geometry
  • Mathematics of computing → Approximation algorithms
Keywords
  • Graph Drawing
  • Chess
  • Hamiltonian Cycle
  • Approximation Algorithms

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References

  1. Karla Alwan and Kelly Waters. Finding re-entrant knight’s tours on n-by-m boards. In Proceedings of the 30th Annual Southeast Regional Conference, ACM-SE 30, pages 377-382, New York, NY, USA, 1992. ACM. URL: https://doi.org/10.1145/503720.503806.
  2. W.W. Rouse Ball and H.S.M. Coxeter. Mathmatical Recreations & Essays: 12th Edition. University of Toronto Press, 1974. URL: http://www.jstor.org/stable/10.3138/j.ctt15jjcrn.
  3. John D. Beasley. Magic knight’s tours. The College Mathematics Journal, 43(1):72-75, 2012. URL: http://www.jstor.org/stable/10.4169/college.math.j.43.1.072.
  4. Ernest Bergholt. Three memoirs on knight’s tours. The Games and Puzzles Journal, 2(18):327-341, 2001. Google Scholar
  5. G.L. Chia and Siew-Hui Ong. Generalized knight’s tours on rectangular chessboards. Discrete Applied Mathematics, 150(1):80-98, 2005. URL: https://doi.org/10.1016/j.dam.2004.11.008.
  6. Axel Conrad, Tanja Hindrichs, Hussein Morsy, and Ingo Wegener. Solution of the knight’s hamiltonian path problem on chessboards. Discrete Applied Mathematics, 50(2):125-134, 1994. URL: https://doi.org/10.1016/0166-218X(92)00170-Q.
  7. Paul Cull and Jeffery De Curtins. Knight’s tour revisited. Fibonacci Quarterly, 16:276-285, June 1978. Google Scholar
  8. Italo J. Dejter. Equivalent conditions for euler’s problem on z₄-hamilton cycles. Ars Combinatoria, 16-B:285-295, 1983. Google Scholar
  9. Joe DeMaio. Which chessboards have a closed knight’s tour within the cube? the electronic journal of combinatorics, 14(1):32, 2007. Google Scholar
  10. Joe DeMaio and Mathew Bindia. Which chessboards have a closed knight’s tour within the rectangular prism? the electronic journal of combinatorics, 18(1):14, 2011. Google Scholar
  11. Joshua Erde, Bruno Golénia, and Sylvain Golénia. The closed knight tour problem in higher dimensions. the electronic journal of combinatorics, 19(4):9, 2012. Google Scholar
  12. Alexander Fischer. New records in nonintersecting knight paths. The Games and Puzzles Journal, 2006. Google Scholar
  13. Ivan Herman, Guy Melançon, and M. Scott Marshall. Graph visualization and navigation in information visualization: A survey. IEEE Transactions on visualization and computer graphics, 6(1):24-43, 2000. Google Scholar
  14. George P. Jelliss. Non-intersecting paths by leapers. The Games and Puzzles Journal, 2(17):305-310, 1999. Google Scholar
  15. George P. Jelliss. Symmetry in knight’s tours. The Games and Puzzles Journal, 2(16):282-287, 1999. Google Scholar
  16. Nina Kamčev. Generalised knight’s tours. the electronic journal of combinatorics, 21(1):32, 2011. Google Scholar
  17. Richard M. Karp. A characterization of the minimum cycle mean in a digraph. Discrete mathematics, 23(3):309-311, 1978. Google Scholar
  18. Donald E. Knuth. Leaper graphs. The Mathematical Gazette, 78(483):274-297, 1994. URL: http://www.jstor.org/stable/3620202.
  19. Awani Kumar. Non-crossing Knight’s Tour in 3-Dimension. ArXiv e-prints, March 2008. URL: http://arxiv.org/abs/0803.4259.
  20. Olaf Kyek, Ian Parberry, and Ingo Wegener. Bounds on the number of knight’s tours. Discrete Applied Mathematics, 74(2):171-181, 1997. URL: https://doi.org/10.1016/S0166-218X(96)00031-5.
  21. Shun-Shii Lin and Chung-Liang Wei. Optimal algorithms for constructing knight’s tours on arbitrary n× m chessboards. Discrete Applied Mathematics, 146(3):219-232, 2005. URL: https://doi.org/10.1016/j.dam.2004.11.002.
  22. Stephen R. Mahaney. Sparse complete sets for np: Solution of a conjecture of berman and hartmanis. Journal of Computer and System Sciences, 25(2):130-143, 1982. URL: https://doi.org/10.1016/0022-0000(82)90002-2.
  23. Brendan D. McKay. Knight’s tours of an 8× 8 chessboard. Technical report, Australian National University, Department of Computer Science, February 1997. Google Scholar
  24. Crispin Nash-Williams. Abelian groups, graphs and generalized knights. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 55(3), pages 232-238. Cambridge University Press, 1959. Google Scholar
  25. Ian Parberry. Scalability of a neural network for the knight’s tour problem. Neurocomputing, 12(1):19-33, 1996. URL: https://doi.org/10.1016/0925-2312(95)00027-5.
  26. Ian Parberry. An efficient algorithm for the knight’s tour problem. Discrete Applied Mathematics, 73(3):251-260, 1997. Google Scholar
  27. Ira Pohl. A method for finding hamilton paths and knight’s tours. Commun. ACM, 10(7):446-449, July 1967. URL: https://doi.org/10.1145/363427.363463.
  28. Yulan Qing and John J. Watkins. Knight’s tours for cubes and boxes. Congressus Numerantium, January 2006. Google Scholar
  29. Allen J. Schwenk. Which rectangular chessboards have a knight’s tour? Mathematics Magazine, 64(5):325-332, 1991. Google Scholar
  30. Jefferey A. Shufelt and Hans J. Berliner. Generating knight’s tours without backtracking from errors. Technical report, Carnegie-Mellon University, School of Computer Science, 1993. Google Scholar
  31. Douglas Squirrel and Paul Cull. A warnsdorff-rule algorithm for knight’s tours on square chessboards, 1996. Google Scholar
  32. John J. Watkins. Across the Board: The Mathematics of Chessboard Problems. Princeton Puzzlers. Princeton University Press; Reissue edition, 2012. Google Scholar
  33. John J. Watkins and Rebecca L. Hoenigman. Knight’s tours on a torus. Mathematics Magazine, 70(3):175-184, 1997. URL: https://doi.org/10.1080/0025570X.1997.11996528.
  34. L. D. Yarbrough. Uncrossed knight’s tours. Journal of Recreational Mathematics, 1(3):140-142, 1969. Google Scholar
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