Deletion in abstract Voronoi diagrams in expected linear time and related problems

TL;DR

This paper presents an expected linear-time algorithm for updating abstract Voronoi diagrams after deletion of a site, introduces the concept of Voronoi-like diagrams, and extends the approach to related structures.

Contribution

It introduces Voronoi-like diagrams as a robust, incremental structure and provides a non-standard analysis for their use in efficient updates and related problems.

Findings

Expected linear-time update algorithm for abstract Voronoi diagrams.

Introduction of Voronoi-like diagrams as a new, robust construct.

Extension of the method to compute order-(k+1) subdivisions and farthest diagrams.

Abstract

Updating an abstract Voronoi diagram after deletion of one site in linear time has been a well-known open problem; similarly, for concrete Voronoi diagrams of non-point sites. In this paper, we present an expected linear-time algorithm to update an abstract Voronoi diagram after deletion of one site. We introduce the concept of a Voronoi-like diagram, a relaxed version of an abstract Voronoi construct that has a structure similar to an ordinary Voronoi diagram, without, however, being one. We formalize the concept, and prove that it is robust under insertion, therefore, enabling its use in incremental constructions. The time-complexity analysis of the resulting simple randomized incremental construction is non-standard, and interesting in its own right, because the intermediate Voronoi-like structures are order-dependent. We further extend the approach to compute the following structures 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.