An almost optimal algorithm for Voronoi diagrams of non-disjoint line segments

This paper presents an almost optimal algorithm that computes the Voronoi diagram of a set S of n line segments that may intersect or cross each other. If there are k intersections among the input segments in S, our algorithm takes O(nα(n)log⁡n+k) time, where α(⋅) denotes the inverse of the Ackerman...

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Bibliographic Details
Published in:	Computational geometry : theory and applications Vol. 52; pp. 34 - 43
Main Author:	Bae, Sang Won
Format:	Journal Article
Language:	English
Published:	Elsevier B.V 01.02.2016
Subjects:	Line segments Medial axis Polygon with holes Voronoi diagram Weakly simple polygon Weakly simple polygon Voronoi diagram Line segments Medial axis Polygon with holes
ISSN:	0925-7721
Online Access:	Get full text
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Summary:	This paper presents an almost optimal algorithm that computes the Voronoi diagram of a set S of n line segments that may intersect or cross each other. If there are k intersections among the input segments in S, our algorithm takes O(nα(n)log⁡n+k) time, where α(⋅) denotes the inverse of the Ackermann function. The best known running time prior to this work was O((n+k)log⁡n). Since the lower bound of the problem is shown to be Ω(nlog⁡n+k) in the worst case, our algorithm is worst-case optimal for k=Ω(nα(n)log⁡n), and is only a factor of α(n) away from any optimal-time algorithm, which is still unknown. For the purpose, we also present an improved algorithm that computes the medial axis or the Voronoi diagram of a polygon with holes.
ISSN:	0925-7721
DOI:	10.1016/j.comgeo.2015.11.002