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.
@InProceedings{besa_et_al:LIPIcs.FUN.2021.4, author = {Besa, Juan Jose and Johnson, Timothy and Mamano, Nil and Osegueda, Martha C.}, title = {{Taming the Knight’s Tour: Minimizing Turns and Crossings}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {4:1--4:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-145-0}, ISSN = {1868-8969}, year = {2020}, volume = {157}, editor = {Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.4}, URN = {urn:nbn:de:0030-drops-127654}, doi = {10.4230/LIPIcs.FUN.2021.4}, annote = {Keywords: Graph Drawing, Chess, Hamiltonian Cycle, Approximation Algorithms} }
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