Planar $\beta$-skeletons via point location in monotone subdivisions of subset of lunes

TL;DR

This paper introduces a new algorithm for computing lune-based $eta$-skeletons for planar point sets with $eta > 2$, achieving the best known running time by utilizing point location in monotone subdivisions.

Contribution

The paper presents an improved algorithm for lune-based $eta$-skeletons for $eta > 2$, based on point location in monotone subdivisions, filling a gap where optimal algorithms were unknown.

Findings

Achieves $O(n^{3/2} \log^{1/2} n)$ running time, the best known for this problem.

Improves upon previous results by Rao and Mukhopadhyay.

Uses a novel approach based on point location in monotone subdivisions.

Abstract

We present a new algorithm for lune-based $\beta$-skeletons for sets of $n$ points in the plane, for $\beta \in (2,\infty]$, the only case when optimal algorithms are not known. The running time of the algorithm is $O(n^{3/2} \log^{1/2} n)$, which is the best known and is an improvement of Rao and Mukhopadhyay \cite{rm97} result. The method is based on point location in monotonic subdivisions of arrangements of curve segments.