$\beta$-skeletons for a Set of Line Segments in $R^2$

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 $\beta \ge 1$ 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.

This research is supported by the ESF EUROCORES program EUROGIGA, CRP VORONOI.