# The Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the Sites

@inproceedings{Reem2011TheGS,
title={The Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the Sites},
author={Daniel Reem},
booktitle={SoCG '11},
year={2011}
}
• Daniel Reem
• Published in SoCG '11 21 March 2011
• Mathematics
Voronoi diagrams appear in many areas in science and technology and have numerous applications. They have been the subject of extensive investigation during the last decades. Roughly speaking, they are a certain decomposition of a given space into cells, induced by a distance function and by a tuple of subsets called the generators or the sites. Consider the following question: does a small change of the sites, e.g., of their position or shape, yield a small change in the corresponding Voronoi…
50 Citations

## Figures from this paper

The geometric stability of Voronoi diagrams in normed spaces which are not uniformly convex
This paper shows that under mild conditions Voronoi diagrams have a certain continuity property, assuming the unit sphere of the space has a certain (non-exotic) structure and the sites satisfy a certain "general position" condition related to it.
Zone diagrams in compact subsets of uniformly convex normed spaces
• Mathematics
ArXiv
• 2010
This paper proves the existence of zone diagrams with respect to finitely many pairwise disjoint compact sites contained in a compact and convex subset of a uniformly convex normed space, provided that either the sites or the conveX subset satisfy a certain mild condition.
On the Existence of a Neutral Region
• Daniel Reem
• Mathematics
2012 Ninth International Symposium on Voronoi Diagrams in Science and Engineering
• 2012
This work presents a simple necessary and sufficient condition ensuring the non-existence of a neutral region in a given space and shows that this assertion is true in a wide class of cases, but not in general.
Topological properties of sets represented by an inequality involving distances
This paper discusses this phenomenon assuming the set is a Voronoi cell induced by given sites (subsets), a geometric object which appears in many fields of science and technology and has diverse applications.
Voronoi Diagram and Delaunay Triangulation with Independent and Dependent Geometric Uncertainties
• Computer Science, Mathematics
Int. J. Comput. Geom. Appl.
• 2021
We address the problems of constructing the Voronoi diagram (VD) and Delaunay triangulation (DT) of points in the plane with mutually dependent location uncertainties, testing their stability, and
On the Computation of Zone and Double Zone Diagrams
• Daniel Reem
• Mathematics, Computer Science
Discret. Comput. Geom.
• 2018
The possibility to compute zone diagrams in a wide class of spaces is discussed and a generalization of the iterative method suggested by Asano, Matoušek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general.
Persistent Homology as Stopping-Criterion for Natural Neighbor Interpolation
• Computer Science
ArXiv
• 2019
In this study the method of natural neighbours is used to interpolate data that has been drawn from a topological space with higher homology groups on its filtration to capture the changing topology of the data.
Voronoi cell analysis: The shapes of particle systems
• Physics
ArXiv
• 2022
Many physical systems can be studied as collections of particles embedded in space, evolving through deterministic evolution equations. Natural questions arise concerning how to characterize these
Generalized Conics: Properties and applications
• Computer Science
2020 25th International Conference on Pattern Recognition (ICPR)
• 2021
The properties of the generalized conics are used to create a unified framework for generating various types of the distance fields and provide a possibility to efficiently compute the proximity, arithmetic mean of the distances and a space tessellation with regard to the given set of polygonal objects, line segments and points.

## References

SHOWING 1-10 OF 131 REFERENCES
An Algorithm for Computing Voronoi Diagrams of General Generators in General Normed Spaces
• Daniel Reem
• Computer Science
2009 Sixth International Symposium on Voronoi Diagrams
• 2009
This work presents an efficient and simple algorithm for computing Voronoi diagrams in general normed spaces, possibly infinite dimensional and allows infinitely many generators of a general form.
Stability of voronoi neighborship under perturbations of the sites
Algorithms are proposed for deciding stability with regard to a given perturbation bound and for determining the supremal bound up to which a particular pair of Voronoi neighbors remains stable.
Existence of zone diagrams in compact subsets of uniformly convex spaces
• Mathematics
CCCG
• 2010
The proof is based on the Schauder fixed point theorem, the Curtis-Schori theorem regarding the Hilbert cube, and on recent results concerning the characterization of Voronoi cells as a collection of line segments and their geometric stability with respect to small changes of the corresponding sites.
Voronoi diagrams—a survey of a fundamental geometric data structure
Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science— such as computer
Voronoi Diagrams of Moving Points
• Mathematics, Computer Science
Int. J. Comput. Geom. Appl.
• 1998
This paper presents a method of maintaining the Voronoi diagram over time, at a cost of O (log n) per event, while showing that the number of topological events has an upper bound of O(ndλs(n)), where λs( n) is the (nearly linear) maximum length of a (n,s)-Davenport-Schinzel sequence, and s is a constant depending on the motions of the point sites.
The stability of the Voronoi diagram
• Mathematics
• 1996
The well-known problem of computational geometry of constructing a Voronoi diagram in R d , is considered. The concept of the stability of the solution of this problem with respect to disturbances of
Kinetic stable Delaunay graphs
• Computer Science
SCG
• 2010
The notion of a stable Delaunay graph (SDG in short) is introduced, a dynamic subgraph of the Delauny triangulation that is less volatile in the sense that it undergoes fewer topological changes and yet retains many useful properties of the full Delaunays.
VORONOI DIAGRAMS AND ORNAMENTAL DESIGN
This work presents some techniques for creating attractive ornamental designs using Voronoi diagrams, focusing on their conservation of symmetry, which can be used to construct interesting tilings of the plane, and their continuity with respect to changes in the generators, which makes possible smooth, organic animations of tilings.
Space-efficient approximate Voronoi diagrams
• Computer Science
STOC '02
• 2002
A data structure that can answer approximate nearest neighbor queries in $O(\log n + 1/\epsilon^{(d-1)/2})$ time using optimal $O(n)$ space.
A replacement for Voronoi diagrams of near linear size
• Sariel Har-Peled
• Computer Science
Proceedings 2001 IEEE International Conference on Cluster Computing
• 2001
A new type of space decomposition that provides an /spl epsi/-approximation to the distance function associated with the Voronoi diagram of P, while being of near linear size, for d/spl ges/2.