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Deletion in Abstract Voronoi Diagrams in Expected Linear Time and Related Problems.

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Název: Deletion in Abstract Voronoi Diagrams in Expected Linear Time and Related Problems.
Autoři: Junginger, Kolja, Papadopoulou, Evanthia
Zdroj: Discrete & Computational Geometry; Jun2023, Vol. 69 Issue 4, p1040-1078, 39p
Témata: VORONOI polygons
Abstrakt: Updating an abstract Voronoi diagram in linear time, after deletion of one site, has been an open problem in a long time; similarly, for any concrete Voronoi diagram of generalized (non-point) sites. In this paper we present a simple, expected linear-time algorithm to update an abstract Voronoi diagram after deletion of one site. To achieve this result, we use the concept of a Voronoi-like diagram, a relaxed Voronoi structure of independent interest. Voronoi-like diagrams serve as intermediate structures, which are considerably simpler to compute, thus, making an expected linear-time construction possible. We formalize the concept and prove that it is robust under insertion, therefore, enabling its use in incremental constructions. The time-complexity analysis introduces a variant to backwards analysis, which is applicable to order-dependent structures. We further extend the technique to compute in expected linear time: the order- (k + 1) subdivision within an order-k Voronoi region, and the farthest abstract Voronoi diagram, after the order of its regions at infinity is known. [ABSTRACT FROM AUTHOR]
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Databáze: Complementary Index
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Abstrakt:Updating an abstract Voronoi diagram in linear time, after deletion of one site, has been an open problem in a long time; similarly, for any concrete Voronoi diagram of generalized (non-point) sites. In this paper we present a simple, expected linear-time algorithm to update an abstract Voronoi diagram after deletion of one site. To achieve this result, we use the concept of a Voronoi-like diagram, a relaxed Voronoi structure of independent interest. Voronoi-like diagrams serve as intermediate structures, which are considerably simpler to compute, thus, making an expected linear-time construction possible. We formalize the concept and prove that it is robust under insertion, therefore, enabling its use in incremental constructions. The time-complexity analysis introduces a variant to backwards analysis, which is applicable to order-dependent structures. We further extend the technique to compute in expected linear time: the order- (k + 1) subdivision within an order-k Voronoi region, and the farthest abstract Voronoi diagram, after the order of its regions at infinity is known. [ABSTRACT FROM AUTHOR]
ISSN:01795376
DOI:10.1007/s00454-022-00463-z