Papers › Taming the Knight's Tour: Minimizing Turns and Crossings
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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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×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 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(n²/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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