Output-Sensitive Algorithm for Computing β-Skeletons

Abstract The β-skeleton is a measure of the internal shape of a planar set of points. We get an entire spectrum of shapes by varying the parameter β. For a fixed value of β, a β-skeleton is a geometric graph obtained by joining each pair of points whose β-neighborhood is empty.For β≥1, this neighborhood of a pair of points pi,pj is the interior of the intersection of two circles of radius , centered at the points (1−β/2)pi+(β/2)pj and (β/2)pi+(1−β/2)pj, respectively. For β∈(0,1], it is the interior of the intersection of two circles of radius , passing through pi and pj.In this paper we present an output-sensitive algorithm for computing a β-skeleton in the metrics l1 and l∞ for any β≥2. This algorithm is in O(nlogn+k), where k is size of the output graph. The complexity of the previous best known algorithm is in O(n5/2logn) [7].