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arXiv · 1904.02824

Taming the Knight's Tour: Minimizing Turns and Crossings

Juan Jose Besa·Timothy Johnson·Nil Mamano·Martha C. Osegueda·Parker Williams
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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 $9.25n+O(1)$ turns and $12n+O(1)$ crossings on an $n\times n$ board, and we show lower bounds of $(6-\epsilon)n$ and $4n-O(1)$ on the respective problems of minimizing these metrics. Hence, our algorithm achieves approximation ratios of $9.25/6+o(1)$ and $3+o(1)$. Our algorithm takes linear time and is fully parallelizable, i.e., the tour can be computed in $O(n^2/p)$ time using $p$ processors in the CREW PRAM model. 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.

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Juan Jose Besa, Timothy Johnson, Nil Mamano, Martha C. Osegueda, Parker Williams. 2019-04-04. Taming the Knight's Tour: Minimizing Turns and Crossings. https://arxiv.org/abs/1904.02824

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