Consider a set of n points in d-dimensional Euclidean space, d 2, each of which i s continuously moving along a given individual trajectory. A t e a c h instant in time, the points de ne a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Voronoi diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has an upper bound of O(n d s (n)), where s (n) i s t h e m a x i m um length of a (n s)-Davenport-Schinzel sequence AgShSh 89, DaSc 65] and s is a constant depending on the motions of the point sites. Our results are a linear-factor improvement o ver the naive O(n d+2 ) upper bound on the number of topological events.In addition, we show that if only k points are moving (while leaving the other n ; k points xed), there is an upper bound of O(kn d;1 s (n) + ( n ;k) d s (k)) on the number of topological events.We give a n umerically stable algorithm for the update of the topological structure of the Voronoi diagram, using only O(log n) time per event (which i s w orst-case optimal per event).

, G e r m a n y T h o m a s Roos t D e p a r t e m e n t Informatik E T H Z e n t r u m CH-8092 Z~rich, Switzerland AbstractThe modeling of realistic dynamic scenes often requires the maintenance of geometric data structures over time. This is the subject of a rising discipline called dynamic computational geometry. In the present work we investigate the behavior of spatial Voronoi diagrams under continuous motions of the underlying sites. Nevertheless, the methodology presented can be applied to many other geometric data structures in computational geometry, as well. Now, 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.We classify the appearing events which cause a change in the topology of the Voronoi diagram and present an algorithm for maintaining the Voronoi diagram over time using only O(log n) time per event which is worst-case optimal. In addition, we give an O(n d ~ (n)) upper bound on the number of topological events. Thereby ~, (n) denotes the maximum length of an (n, s)-Davenport-Schinzel sequence, and s is a constant depending on the underlying trajectories of the moving sites. I n t r o d u c t i o nVoronoi diagrams are a fundamental tool expressing the proximity of geometric objects. So, it is not surprising that they appear in many variations in computational geometry as well as other scientific areas related (see [Au 90] for a survey on this topic).A problem of recent interest has been of allowing the set of objects S to vary continuously over time. This "dynamic" version has been studied in the case of points in the Euclidean plane In this paper, we consider the following problem: We are given a set S of n points in d-dimensional Euclidean space, d ~ 3, each of which is continuously moving along a given

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This paper presents a refined layered architecture for business information systems of any size. It allows a strict separation of application logic, database access, and user interface and is largely independent of programming languages, database management systems, operating systems, and middleware.

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