Multidimensional $\beta$-skeletons in $L_1$ and $L_{\infty}$ metric

The β-skeleton {Gβ(V )} for a point set V is a family of geometric graphs, defined by the notion of neighborhoods parameterized by real number 0 < β < ∞. By using the distance-based version definition of β-skeletons we study those graphs for a set of points in R d space with l1 and l∞ metrics. We present algorithms for the entire spectrum of β values and we discuss properties of lens-based and circle-based β-skeletons in those metrics. Let V ∈ R in L∞ metric be a set of n points in general position. Then, for β < 2 lens-based β-skeleton Gβ(V ) can be computed in O(n 2 log n) time. For β ≥ 2 there exists an O(n log n) time algorithm that constructs β-skeleton for the set V . We show that in R with L∞ metric, for β < 2 β-skeleton Gβ(V ) for n points can be computed in O(n 2 log n) time. For β ≥ 2 there exists an O(n log n) time algorithm. In R with L1 metric for a set of n points in arbitrary position β-skeleton Gβ(V ) can be computed in O(n 2 log n) time.