On Deletion in Delaunay Triangulation

TL;DR

This paper introduces an efficient method for deleting points from Delaunay triangulations using sphere space and shelling, correcting a false algorithm previously used in GIS applications.

Contribution

It proposes a novel deletion algorithm based on sphere and shelling concepts, and clarifies errors in the existing Heller's algorithm.

Findings

Deletion complexity is O(k log k) in 2D, where k is the degree of the deleted vertex.

The algorithm's main operations are low cost, with time-consuming steps linear in number.

Heller's algorithm is shown to be incorrect, and this paper provides the correct approach.

Abstract

This paper presents how the space of spheres and shelling may be used to delete a point from a $d$-dimensional triangulation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O(k log k), but we notice that this number only applies to low cost operations, while time consuming computations are only done a linear number of times. This algorithm may be viewed as a variation of Heller's algorithm, which is popular in the geographic information system community. Unfortunately, Heller algorithm is false, as explained in this paper.