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arXiv · 2207.08708

Shortest polygonal chains covering each planar square grid

Marco Ripà
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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).

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Marco Ripà. 2024-04-18. Shortest polygonal chains covering each planar square grid. https://arxiv.org/abs/2207.08708

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