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

TL;DR

This paper extends the concept of $eta$-skeletons from points to line segments in the Euclidean plane, providing algorithms for their computation across all $eta$ values, with applications to moving points.

Contribution

It introduces algorithms for computing $eta$-skeletons for line segments for all $eta$ values, generalizing existing point-based methods and analyzing their computational complexity.

Findings

Algorithms for $eta$-skeletons with $eta eq 1$ have polynomial time complexity.

The $eta=1$ case (Gabriel Graph) can be computed efficiently in $O(n ext{ log } n)$ time.

Relations between $eta$-skeletons, Gabriel Graph, and Delaunay triangulation extend to line segments.

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.