A randomized divide and conquer algorithm for higher-order abstract Voronoi diagrams

Given a set of n sites in the plane, their order-k Voronoi diagram partitions the plane into regions such that all points within one region have the same k nearest sites. The order-k abstract Voronoi diagram offers a unifying framework that represents a wide range of concrete order-k Voronoi diagram...

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Bibliographic Details
Published in:	Computational geometry : theory and applications Vol. 59; pp. 26 - 38
Main Authors:	Bohler, Cecilia, Liu, Chih-Hung, Papadopoulou, Evanthia, Zavershynskyi, Maksym
Format:	Journal Article
Language:	English
Published:	Elsevier B.V 01.12.2016
Subjects:	Abstract Voronoi diagram Divide and conquer Geometric randomized algorithm Higher-order Voronoi diagram k nearest neighbors Geometric randomized algorithm Higher-order Voronoi diagram k nearest neighbors Abstract Voronoi diagram Divide and conquer
ISSN:	0925-7721
Online Access:	Get full text
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Summary:	Given a set of n sites in the plane, their order-k Voronoi diagram partitions the plane into regions such that all points within one region have the same k nearest sites. The order-k abstract Voronoi diagram offers a unifying framework that represents a wide range of concrete order-k Voronoi diagrams. It is defined in terms of bisecting curves satisfying some simple combinatorial properties, rather than the geometric notions of sites and distance. In this paper we develop a randomized divide-and-conquer algorithm to compute the order-k abstract Voronoi diagram in expected O(kn1+ε) operations. For solving small sub-instances in the divide-and-conquer process, we also give two auxiliary algorithms with expected O(k2nlog⁡n) and O(n22α(n)log⁡n) time, respectively, where α(⋅) is the inverse of the Ackermann function. Our approach directly implies an O(kn1+ε)-time algorithm for several concrete order-k instances such as points in any convex distance, disjoint line segments or convex polygons of constant size in the Lp norms, and others. It also provides basic techniques that can enable the application of well-known random sampling techniques to the construction of Voronoi diagrams in the abstract setting and for non-point sites.
ISSN:	0925-7721
DOI:	10.1016/j.comgeo.2016.08.004