Shortest polygonal chains covering each planar square grid
Abstract
Given any $n \in \mathbb{Z}^{+}$, we constructively prove the existence of covering paths and circuits in the plane which are characterized by the same link length of the minimum-link covering trails for the two-dimensional grid $G_n^2 := \{0,1, \ldots, n-1\} \times \{0, 1, \ldots, n-1\}$. Furthermore, we introduce a general algorithm that returns a covering cycle of analogous link length for any even value of $n$. Finally, we provide the tight upper bound $n^2 - 3 + 5 \cdot \sqrt{2}$ units for the minimum total distance travelled to visit all the nodes of $G_n^2$ with a minimum-link trail (i.e., a trail with $2 \cdot n - 2$ edges if $n$ is above two).
- Publication:
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arXiv e-prints
- Pub Date:
- July 2022
- DOI:
- 10.48550/arXiv.2207.08708
- arXiv:
- arXiv:2207.08708
- Bibcode:
- 2022arXiv220708708R
- Keywords:
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- Mathematics - Combinatorics;
- 05C38 (Primary) 05C12;
- 91A43 (Secondary)
- E-Print:
- 18 pages, 13 figures