Taming the Knight's Tour: Minimizing Turns and Crossings

Juan José Besa Vial, Timothy Johnson, Nil Mamano, Martha C. Osegueda
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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 $9.5n+O(1)$ turns and $13n+O(1)$ crossings on a $n\times n$ board. We show lower bounds of $(6-\varepsilon)n$, for any $\varepsilon>0$, and $4n-O(1)$ on the respective problems of minimizing these metrics. Hence, we achieve 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.
驯服骑士之旅:减少转弯和交叉
我们为骑士的旅行引入了两个新的简单度量:转弯次数和交叉次数。我们给出了一种新的算法,该算法在$n\ * n$的棋盘上产生$9.5n+O(1)$次转弯和$13n+O(1)$次交叉的旅行。我们给出了$(6-\varepsilon)n$的下界,对于任何$\varepsilon>0$,以及$4n-O(1)$在最小化这些度量的各自问题上的下界。因此,我们获得了$19/12+o(1)$和$13/4+o(1)$的近似比率。我们将我们的技术推广到矩形棋盘,高维棋盘,对称游,缺角的奇数棋盘,以及$(1,4)$-跳子的游。在此过程中,我们证明了这些扩展在最小匝数和大多数情况下的交叉数上也承认一个常数近似比。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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