General uncrossing covering paths inside the axis-aligned bounding box
Marco Ripà
TL;DR
For any n_k≥3, the link length of the covering path provided by the MΛI-algorithm is smaller than the cardinality of the set G(n_1, n_2,...,n_k).
Abstract
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 $\left[0,4-\sqrt{3}\right] \times \left[0,4-\sqrt{3}\right] \times [0, 2]^{k-2}$.