A constructive approach to sweep algorithms for Voronoi diagrams
Date
1991
Authors
Heuberger, Philipp A. (Philipp AndreĢ)
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Abstract
Constructing the Voronoi diagram where the given sites are points in the Euclidean space veils the discrete problem of determining the finite set of siteless subsets of the original sites. The function siteless incorporates all geometric and the function subset all combinatorial concerns. The set of all siteless subsets is called the Voronoi set. Independent of dimension, our precise definition of the Voronoi diagram is based on the Voronoi set. Thus, the finite Voronoi set is a convenient abstraction of the infinite Voronoi diagram separating geometric from combinatorial concerns. Furthermore, this thesis describes a constructive approach to Voronoi sweep algorithms. In a first attempt, we derive a brute-force algorithm demonstrating our methodology of strengthening program conditions. Then, two efficient algorithms for the plane are presented: the sweephull and the sweepcircle Voronoi algorithm. The sweepcircle algorithm is a companion of Fortune's sweepline algorithm, where the sweepline has been replaced by a sweepcircle. The sweephull algorithm, a generalized form of both sweepline and sweepcircle algorithms, is based on our improved underĀstanding of the relationship between the convex hull and the Voronoi diagram.