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

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 diag...

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Veröffentlicht in:Discrete & computational geometry Jg. 69; H. 4; S. 1040 - 1078
Hauptverfasser: Junginger, Kolja, Papadopoulou, Evanthia
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.06.2023
Springer Nature B.V
Schlagworte:
Algorithms
Combinatorics
Computational Mathematics and Numerical Analysis
Concrete
Construction
Deletion
Geometry
Mathematics
Mathematics and Statistics
Voronoi graphs
Farthest Voronoi diagram
Site-deletion
Higher-order Voronoi diagram
Randomized incremental construction
68W05
68U05
Abstract Voronoi diagram
Linear-time algorithm
Backwards analysis
ISSN:0179-5376, 1432-0444
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Zusammenfassung: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.
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Editor in Charge: Kenneth Clarkson
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-022-00463-z