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

The $\beta$-skeleton $\{G_{\beta}(V)\}$ for a point set V is a family of geometric graphs, defined by the notion of neighborhoods parameterized by real number $0 < \beta < \infty$. By using the distance-based version definition of $\beta$-skeletons we study those graphs for a set of points in $\mathbb{R}^d$ space with $l_1$ and $l_{\infty}$ metrics. We present algorithms for the entire spectrum of $\beta$ values and we discuss properties of lens-based and circle-based $\beta$-skeletons in those metrics. Let $V \in \mathbb{R}^d$ in $L_{\infty}$ metric be a set of $n$ points in general position. Then, for $\beta<2$ lens-based $\beta$-skeleton $G_{\beta}(V)$ can be computed in $O(n^2 \log^d n)$ time. For $\beta \geq 2$ there exists an $O(n \log^{d-1} n)$ time algorithm that constructs $\beta$-skeleton for the set $V$. We show that in $\mathbb{R}^d$ with $L_{\infty}$ metric, for $\beta<2$ $\beta$-skeleton $G_{\beta}(V)$ for $n$ points can be computed in $O(n^2 \log^d n)$ time. For $\beta \geq 2$ there exists an $O(n \log^{d-1} n)$ time algorithm. In $\mathbb{R}^d$ with $L_1$ metric for a set of $n$ points in arbitrary position $\beta$-skeleton $G_{\beta}(V)$ can be computed in $O(n^2 \log^{d+2} n)$ time.