$\beta$-skeletons for a set of line segments in $R^2$

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Abstract

$\beta$-skeletons are well-known neighborhood graphs for a set of points. We extend this notion to sets of line segments in the Euclidean plane and present algorithms computing such skeletons for the entire range of $\beta$ values. The main reason of such extension is the possibility to study $\beta$-skeletons for points moving along given line segments. We show that relations between $\beta$-skeletons for $\beta > 1$, $1$-skeleton (Gabriel Graph), and the Delaunay triangulation for sets of points hold also for sets of segments. We present algorithms for computing circle-based and lune-based $\beta$-skeletons. We describe an algorithm that for $\beta \geq 1$ computes the $\beta$-skeleton for a set $S$ of $n$ segments in the Euclidean plane in $O(n^2 \alpha (n) \log n)$ time in the circle-based case and in $O(n^2 \lambda_4(n))$ in the lune-based one, where the construction relies on the Delaunay triangulation for $S$, $\alpha$ is a functional inverse of Ackermann function and $\lambda_4(n)$ denotes the maximum possible length of a $(n,4)$ Davenport-Schinzel sequence. When $0 < \beta < 1$, the $\beta$-skeleton can be constructed in a $O(n^3 \lambda_4(n))$ time. In the special case of $\beta = 1$, which is a generalization of Gabriel Graph, the construction can be carried out in a $O(n \log n)$ time.