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Maintaining proximity in higher dimensional spaces

Extended abstract
  • Thomas Roos
  • Gerhard Albers
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 629)

Abstract

Dynamic computational geometry is concerned with the maintenance of geometric data structures over time. In the present work we investigate the behavior of spatial nearestneighbor Voronoi diagrams and planar higher-order Voronoi diagrams under continuous motions of the underlying sites. Nevertheless, the methodology presented is interesting in itself and can be applied to many other geometric data structures in computational geometry, as well.

In this paper, we consider a set of n points moving continuously along given trajectories in d-dimensional Euclidean space, d≥3. At each instant, the points define a Voronoi diagram which changes continuously except of certain critical instants, so-called topological events. At first, we classify the events which appear in the d-dimensional nearest-neighbor Voronoi diagram and outline an O(n d λ s (n)) upper bound on the number of topological events. Thereby λ s (n) denotes the maximum length of a (n, s)-Davenport-Schinzel sequence, and s is a constant depending on the underlying trajectories of the moving sites.

Next, we apply our methodology to planar order-m Voronoi diagrams, m- 1,..., n−1. The basic result in this context is that topological events affect at most three successive orders of Voronoi diagrams. Additionally, we give a tight O(n 4) upper bound on the number of topological events which appear in all order-m Voronoi diagrams.

Both cases allow simple and numerically stable algorithms for maintaining the corresponding Voronoi diagram(s) over time using only O(log n) time per event, which is worstcase optimal. This work generalizes the most recent results by [FuLe 91], [GuMiRo 91] and [ImIm 90] to three and higher dimensions as well as higher orders.

Keywords

Topological Structure Voronoi Diagram Computational Geometry Dual Graph Topological Event 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Thomas Roos
    • 1
  • Gerhard Albers
    • 2
  1. 1.Departement InformatikETH ZentrumZürichSwitzerland
  2. 2.Lehrstuhl für InformatikUniversität WürzburgWürzburgGermany

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