Fast algorithms for computing Beta-skeletons and their relatives

Abstract In this paper we present fast algorithms for computing β-skeletons (Kirkpatrick and Radke, in: Toussaint (Ed.), Computational Geometry, North-Holland, Amsterdam, 1985, pp. 217–248) and two of its relatives, namely, kβ-skeletons, and additively weighted β-skeletons. A β-skeleton is a generalization of the Relative Neighborhood Graph, introduced by Toussaint (Toussaint, Pattern Recognition 12 (1980) 261–268). Our algorithms are in O(n 3/2 log n) for β⩾1 and in O(n 5/2 log n) for β∈[0, 1) under the metric Lp for 1 O(n 5/2 log n) . Given the Delaunay triangulation, the linear time algorithms known todate for computing β-skeleton are restricted to the range 1⩽β⩽2 under the metric Lp for 1

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