Deletion in Abstract Voronoi Diagrams in Expected Linear Time and Related Problems - Junginger, Kolja; Papadopoulou, Evanthia (Discrete & computational geometry, 2023)
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.