Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Zone Diagrams: Existence, Uniqueness, and Algorithmic Challenge
SIAM Journal on Computing
Some theoretical challenges in digital geometry: A perspective
Discrete Applied Mathematics
Voronoi Diagrams and Polynomial Root-Finding
ISVD '09 Proceedings of the 2009 Sixth International Symposium on Voronoi Diagrams
An Algorithm for Computing Voronoi Diagrams of General Generators in General Normed Spaces
ISVD '09 Proceedings of the 2009 Sixth International Symposium on Voronoi Diagrams
Maximal Zone Diagrams and their Computation
ISVD '10 Proceedings of the 2010 International Symposium on Voronoi Diagrams in Science and Engineering
Polynomial Root-Finding Methods Whose Basins of Attraction Approximate Voronoi Diagram
Discrete & Computational Geometry
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The notion of the zone diagram of a finite set of points in the Euclidean plane is an interesting and rich variation of the classical Voronoi diagram, introduced by Asano, Matoušek, and Tokuyama [1]. In this paper, we define mollified versions of zone diagram named territory diagram and maximal territory diagram. A zone diagram is a particular maximal territory diagram satisfying a sharp dominance property. The proof of existence of maximal territory diagrams depends on less restrictive initial conditions and is established via Zorn's lemma in contrast to the use of fixed-point theory in proving the existence of the zone diagram. Our proof of existence relies on a characterization which allows embedding any territory diagram into a maximal one. Our analysis of the structure of maximal territory diagrams is based on the introduction of a pair of dual concepts we call safe zone and forbidden zone. These in turn give rise to computational algorithms for the approximation of maximal territory diagrams. Maximal territory diagrams offer their own interesting theoretical and computational challenges, as well as insights into the structure of zone diagrams. This paper extends and updates previous work presented in [4].