Marco Ripà In the Euclidean space \(\mathbb{R}^3\), we ask whether one can visit each of the \(27\) vertices of the grid \(G_3:=\{0,1,2\}^3\) exactly once using as few straight-line segments, connected end to end, as possible (an optimal polygonal chain). We give a constructive proof that there exists a \(13\)-segment perfect simple path (i.e., an optimal chain that … Read more
A generalization of Ripà’s square spiral solution for the n × n × ··· × n Points Upper Bound Problem. Additionally, we provide a non-trivial lower bound for the k-dimensional n_1 × n_2 × ··· × n_k Points Problem. In this way, we can build a range in which, with certainty, all the best possible … Read more
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, … Read more
In this paper, we show enhanced upper bounds of the nontrivial \(n_1 \times n_2 \times n_3\) points problem for every \(n_1 \leq n_2 \leq n_3 < 6\). We present new patterns that significantly improve the previously known algorithms for finding minimum-link covering paths and trails. CitationAn old version of the present paper has been published … Read more