General uncrossing covering paths inside the axis-aligned bounding box
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909089885323264 |
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| author | Ripà, Marco |
| author_facet | Ripà, Marco |
| contents | Given the finite set of $n_1 \cdot n_2 \cdot \ldots \cdot n_k$ points $G_{n_1,n_2,\ldots,n_k} \subset \mathbb{R}^k$ such that $n_k \geq \cdots \geq n_2 \geq n_1 \in \mathbb{Z}^+$, we introduce a new algorithm, called M$Λ$I, which returns an uncrossing covering path inside the minimum axis-aligned bounding box $[0,n_1-1] \times [0,n_2-1] \times \cdots \times [0,n_k-1]$, consisting of $3 \cdot \prod_{i=1}^{k-1} n_i-2$ links of prescribed length $n_k-1$ units. Thus, for any $n_k \geq 3$, the link length of the covering path provided by our M$Λ$I-algorithm is smaller than the cardinality of the set $G_{n_1,n_2,\ldots,n_k}$. Furthermore, assuming $k>2$, we present an uncrossing covering path for $G_{3,3,\ldots,3}$, consisting of $20 \cdot 3^{k-3}-2$ straight-line edges that are $2$ units long each, which is constrained by the axis-aligned bounding box $\left[0,4-\sqrt{3}\right] \times \left[0,4-\sqrt{3}\right] \times [0, 2]^{k-2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00096 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | General uncrossing covering paths inside the axis-aligned bounding box Ripà, Marco Combinatorics 52C17 (Primary) 05C12, 68R10 (Secondary) Given the finite set of $n_1 \cdot n_2 \cdot \ldots \cdot n_k$ points $G_{n_1,n_2,\ldots,n_k} \subset \mathbb{R}^k$ such that $n_k \geq \cdots \geq n_2 \geq n_1 \in \mathbb{Z}^+$, we introduce a new algorithm, called M$Λ$I, which returns an uncrossing covering path inside the minimum axis-aligned bounding box $[0,n_1-1] \times [0,n_2-1] \times \cdots \times [0,n_k-1]$, consisting of $3 \cdot \prod_{i=1}^{k-1} n_i-2$ links of prescribed length $n_k-1$ units. Thus, for any $n_k \geq 3$, the link length of the covering path provided by our M$Λ$I-algorithm is smaller than the cardinality of the set $G_{n_1,n_2,\ldots,n_k}$. Furthermore, assuming $k>2$, we present an uncrossing covering path for $G_{3,3,\ldots,3}$, consisting of $20 \cdot 3^{k-3}-2$ straight-line edges that are $2$ units long each, which is constrained by the axis-aligned bounding box $\left[0,4-\sqrt{3}\right] \times \left[0,4-\sqrt{3}\right] \times [0, 2]^{k-2}$. |
| title | General uncrossing covering paths inside the axis-aligned bounding box |
| topic | Combinatorics 52C17 (Primary) 05C12, 68R10 (Secondary) |
| url | https://arxiv.org/abs/2402.00096 |