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

Abstract

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.