Consider a set of n points in d-dimensional Euclidean space, d 2, each of which is continuously moving along a given individual trajectory. At each instant in time, the points define 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) is the maximum 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 over 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 \Gamma k points fixed), there is an upper bound of O(kn d\Gamma1 s (n) + (n \Gamma k)...

Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is often useful or required to compute what one might call the "shape" of the set. For that purpose, this thesis deals with the formal notion of the family of alpha shapes of a finite point set in three- dimensional space. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a real parameter controlling the desired level of detail. Algorithms and data structures are presented that construct and store the entire family of shapes, with a quadratic time and space complexity, in the worst case.

Understanding the process of categorization is a primary research goal in artificial intelligence. The conceptual space framework provides a flexible approach to modeling context-sensitive categorization via a geometrical representation designed for modeling and managing concepts. In this paper we show how algorithms developed in computational geometry, and the Region Connection Calculus can be used to model important aspects of categorization in conceptual spaces. In particular, we demonstrate the feasibility of using existing geometric algorithms to build and manage categories in conceptual spaces, and we show how the Region Connection Calculus can be used to reason about categories and other conceptual regions. 1

The distance transform (DT) is a general operator forming the basis of many methods in computer vision and geometry, with great potential for practical applications. However, all the optimal algorithms for the computation of the exact Euclidean DT (EDT) were proposed only since the 1990s. In this work, state-of-theart sequential 2D EDT algorithms are reviewed and compared, in an effort to reach more solid conclusions regarding their differences in speed and their exactness. Six of the best algorithms were fully implemented and compared in practice.

The bisector systems of convex distance functions in 3-space are investigated and it is shown that there is a substantial difference to the Euclidean metric which cannot be observed in 2-space. This disproves the general belief that Voronoi diagrams in convex distance functions are, in any dimension, analogous to Euclidean Voronoi diagrams. The fact is that more spheres than one can pass through four points in general position. In the L 4 -metric, there exist quadrupels of points that lie on the surface of three L 4 -spheres. Moreover, for each n # 0 one can construct a smooth and symmetric convex distance function d and four points that are contained in the surface of exactly 2n +1 d-spheres, and this number does not decrease if the four points are disturbed independently within 3-dimensional neighborhoods. This result implies that there is no general upper bound to the complexity of the Voronoi diagram of four sites based on a convex distance function in 3-space.

This paper presents Detri 2.2, an implementation for Delaunay triangulations of three-dimensional point sets. The code uses a variant of the randomized incremental-flip algorithm, and employs a symbolic perturbation scheme to achieve robustness. The algorithm's time complexity is quadratic in n, the number of input points, and this is optimal in the worst case. However, empirically, we can expect running times roughly proportional to the number of triangles in the final triangulation, which typically is linear in n. Even though the symbolic perturbation scheme relies on exact arithmetic, the resulting code is efficient in practice. This is mainly due to a careful implementation of the geometric primitives and the arithmetic module. The source code is freely available on the Internet.

We show that the wavefront approach to Voronoi diagrams (a deterministic line sweep algorithm that does not use geometric transform) can be generalized to distance measures more general than the Euclidean metric. In fact, we provide the first worstcase optimal (O(n log time, O(n) space) algorithm that is valid for the full class of what has been called nice metrics in the plane. This also solves the previously open problem of providing an time plane-sweep algorithm for arbitrary L k -metrics.

The construction of the convex hull of a finite point set in a low-dimensional Euclidean space is a fundamental problem in computational geometry. This thesis investigates efficient algorithms for the convex hull problem, where complexity is measured as a function of both the size of the input point set and the size of the output polytope. Two new, simple, optimal, output-sensitive algorithms are presented in two dimensions and a simple, optimal, output-sensitive algorithm is presented in three dimensions. In four dimensions, we give the first output-sensitive algorithm that is within a polylogarithmic factor of optimal. In higher fixed dimensions, we obtain an algorithm that is optimal for sufficiently small output sizes and is faster than previous methods for sublinear output sizes; this result is further improved in even dimensions. Although the focus of the thesis is on the convex hull problem, applications of our techniques to many related problems in computational geometry are al...

We are given a set of n points moving continuously along given trajectories in d-dimensional Euclidean space. At each instant, these sites define a Voronoi diagram which changes continuously over time except of certain critical instances, so-called topological events [4]. In this paper, we present an algorithm for maintaining the Voronoi diagram in parallel over time using only O(1) time per event on a CREW PRAM with O(n d d 2 e ) processors which is worst-case optimal. This work generalizes the most recent single processor algorithms by [17, 20, 28] to PRAMs. Additionally, the presented technique provides the first efficient algorithm for computing static higher dimensional Voronoi diagrams in parallel. Finally, we present a new upper bound on the number of topological events which may appear during the flow of the sites. This result tightens a long open gap between the upper and lower worst-case bounds actually providing matching bounds in some cases and is interesting in its ow...

over relational databases is to find the minimum Steiner trees in database graphs. These methods, however, are rather expensive as the minimum Steiner tree problem is known to be NP-hard. Further, these methods cannot benefit from DBMS capabilities. We propose a new concept called Compact Steiner Tree (CSTree), which can be used to approximate the Steiner tree problem for answering top-k keyword queries efficiently. We propose a structure-aware index, together with an effective ranking mechanism for fast, progressive and accurate retrieval of top-k highest ranked CSTrees. The proposed techniques can be implemented using a standard RDBMS to benefit from its indexing and query processing capability. The experimental results show that our method achieves high search efficiency and result quality comparing to existing state-of-the-art approaches.