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Taming the Knight’s Tour: Minimizing Turns and Crossings

Authors Juan Jose Besa , Timothy Johnson, Nil Mamano , Martha C. Osegueda

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Author Details

Juan Jose Besa
  • University of California, Irvine, CA, USA
Timothy Johnson
  • University of California, Irvine, CA, USA
Nil Mamano
  • University of California, Irvine, CA, USA
Martha C. Osegueda
  • University of California, Irvine, CA, USA

Funding

The authors were supported by NSF Grant CCF-1616248 and NSF Grant 1815073.

Cite As Get BibTex

Juan Jose Besa, Timothy Johnson, Nil Mamano, and Martha C. Osegueda. Taming the Knight’s Tour: Minimizing Turns and Crossings. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.FUN.2021.4

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× 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.

Subject Classification

ACM Subject Classification
  • Human-centered computing → Graph drawings
  • Theory of computation → Computational geometry
  • Mathematics of computing → Approximation algorithms
Keywords
  • Graph Drawing
  • Chess
  • Hamiltonian Cycle
  • Approximation Algorithms

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References

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