Abstract
We study two aspects of a generalization of the Tower of Hanoi puzzle. In 1981, D. Wood suggested its variant, where a bigger disk may be placed higher than a smaller one if their size difference is less than k. In 1992, D. Poole suggested a natural disk-moving strategy for this problem, but only in 2005, the authors proved it be optimal in the general case. We describe the family of all optimal solutions to this problem and present a closed formula for their number, as a function of the number of disks and k. Besides, we prove a tight bound for the diameter of the configuration graph of the problem suggested by Wood. Finally, we prove that the average length of shortest sequence of moves, over all pairs of initial and final configurations, is the same as the above diameter, up to a constant factor.
Partially supported by the Lynn and William Frankel Center for Computer Science.
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References
Beneditkis, S., Safro, I.: Generalizations of the Tower of Hanoi Problem. Final Project Report, supervised by Berend, D., Dept. of Mathematics and Computer Science, Ben-Gurion University (1998)
Chen, X., Tian, B., Wang, L.: Santa Claus’ Towers of Hanoi. Manuscript (2005)
Dinitz, Y., Solomon, S.: Optimal algorithms for tower of hanoi problems with relaxed placement rules. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 36–47. Springer, Heidelberg (2006)
Poole, D.: The Bottleneck Towers of Hanoi Problem. J. of Recreational Math. 24(3), 203–207 (1992)
Szegedy, M.: In how many steps the k peg version of the towers of hanoi game can be solved? In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, p. 356. Springer, Heidelberg (1999)
Wood, D.: The Towers of Brahma and Hanoi Revisited. J. of Recreational Math. 14(1), 17–24 (1981)
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Dinitz, Y., Solomon, S. (2007). On Optimal Solutions for the Bottleneck Tower of Hanoi Problem. In: van Leeuwen, J., Italiano, G.F., van der Hoek, W., Meinel, C., Sack, H., Plášil, F. (eds) SOFSEM 2007: Theory and Practice of Computer Science. SOFSEM 2007. Lecture Notes in Computer Science, vol 4362. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69507-3_20
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DOI: https://doi.org/10.1007/978-3-540-69507-3_20
Publisher Name: Springer, Berlin, Heidelberg
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