General Uncrossing Covering Paths Inside the Axis-Aligned Bounding Box

Abstract

Given the finite set of n_1⋅n_2⋅...⋅n_k points G(n_1,n_2,...,n_k) in R^𝑘 such that n_k≥...≥n_2≥n_1∈Z+, we introduce a new algorithm, called MΛI, which returns an uncrossing covering path inside the minimum axis-aligned bounding box [0,n_1−1]×[0,n_2−1]×...×[0,n_k−1], consisting of 3⋅(n_1⋅n_2⋅...⋅n_k−1)−2 links of prescribed length n_k−1 units. Thus, for any n_k≥3, the link length of the covering path provided by our MΛI-algorithm is smaller than the cardinality of the set G(n_1,n_2,...,n_k). Furthermore, assuming k>2, we present an uncrossing covering path for G(3,3,...,3), comprising only 20*3^(k−3)−2 two units long edges, which is constrained by the axis-aligned bounding box [0,4−√3]×[0,4−√3]×[0,2]×...×[0,2].