Juan José Besa Vial, Timothy Johnson, Nil Mamano, Martha C. Osegueda
{"title":"Taming the Knight's Tour: Minimizing Turns and Crossings","authors":"Juan José Besa Vial, Timothy Johnson, Nil Mamano, Martha C. Osegueda","doi":"10.4230/LIPIcs.FUN.2021.4","DOIUrl":null,"url":null,"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.","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"64 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fun with Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.FUN.2021.4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.