Minimum-Link Covering Trails for any Hypercubic Lattice
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915444802191360 |
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| author | Ripà, Marco |
| author_facet | Ripà, Marco |
| contents | In 1994, Kranakis et al. published a conjecture about the minimum link-length of every rectilinear covering path for the $k$-dimensional grid $P(n,k) := \{0,1, \dots, n-1\} \times \{0,1, \dots, n-1\} \times \cdots \times \{0,1, \dots, n-1\}$. In this paper, we consider the general, NP-complete, Line-Cover problem, where the edges are not required to be axis-parallel, showing that the original Theorem 1 by Kranakis et al. no longer holds when the aforementioned constraint is disregarded. Furthermore, for any $n$ greater than two, as $k$ approaches infinity, the link-length of any minimal (non-rectilinear) polygonal chain does not exceed Kranakis' conjectured value of $\frac{k}{k-1} \cdot n^{k-1}+O(n^{k-2})$ only if we introduce a multiplicative constant $c \geq 1.5$ for the lower order terms (e.g., if we select $n=3$ and assume that $c<1.5$, starting from a sufficiently large $k$, it is not possible to visit all the nodes of $P(n,k)$ with a trail of link-length $\frac{k}{k-1} \cdot n^{k-1}+c \cdot n^{k-2}$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_01699 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Minimum-Link Covering Trails for any Hypercubic Lattice Ripà, Marco General Mathematics 05C57, 05C38 In 1994, Kranakis et al. published a conjecture about the minimum link-length of every rectilinear covering path for the $k$-dimensional grid $P(n,k) := \{0,1, \dots, n-1\} \times \{0,1, \dots, n-1\} \times \cdots \times \{0,1, \dots, n-1\}$. In this paper, we consider the general, NP-complete, Line-Cover problem, where the edges are not required to be axis-parallel, showing that the original Theorem 1 by Kranakis et al. no longer holds when the aforementioned constraint is disregarded. Furthermore, for any $n$ greater than two, as $k$ approaches infinity, the link-length of any minimal (non-rectilinear) polygonal chain does not exceed Kranakis' conjectured value of $\frac{k}{k-1} \cdot n^{k-1}+O(n^{k-2})$ only if we introduce a multiplicative constant $c \geq 1.5$ for the lower order terms (e.g., if we select $n=3$ and assume that $c<1.5$, starting from a sufficiently large $k$, it is not possible to visit all the nodes of $P(n,k)$ with a trail of link-length $\frac{k}{k-1} \cdot n^{k-1}+c \cdot n^{k-2}$). |
| title | Minimum-Link Covering Trails for any Hypercubic Lattice |
| topic | General Mathematics 05C57, 05C38 |
| url | https://arxiv.org/abs/2208.01699 |