Thomas Roos

Space-filling curves and their use in the design of geometric data structures

We are given a two-dimensional square grid of size N×N, where N∶=2n and n≥0. A space filling curve (SFC) is a numbering of the cells of this grid with numbers from c+1 to c+N2, for some c≥0. We call a SFC recursive (RSFC) if it can be recursively divided into four square RSFCs of equal size. Examples of well-known RSFCs include the Hilbert curve, the z-curve, and the Gray code.

Algorithmic foundations of geographic information systems

to geometric computing: From algorithms to software.- Voronoi methods in GIS.- Digital elevation models and TIN algorithms.- Visualization of TINs.- Generalization of spatial data: Principles and selected algorithms.- Spatial data structures: Concepts and design choices.- Space filling curves versus random walks.- External-memory algorithms with applications in GIS.- Precision and robustness in geometric computations.

k -Violation linear programming

We introduce the notion of k-violation linear programming. Given a set of n halfplanes, we want to compute an optimal solution with respect to a given linear functional. However, in opposite to classical linear programming [MaShWe 92], we allow to violate at most k of the n constraints, for some fixed k e {0,..., n − 1}. We solve this problem in O(β κ (n)) time and O(n) space, where β κ(n) := n log n + κ log2 κ. This is optimal if κ e O(n α) for any fixed positive α<1.

Space Filling Curves versus Random Walks

Data structures for maintaining sets of multidimensional points on external storage are very usefill to non-standard database systems. In contrast to the onedimensional case, the situation in higher dimensions is far more complex, since there is no obvious total order on the points that serves all purposes. The most frequent type of queries on these multidimensional point sets are range queries: Given the set of points and a multidimensional query interval, report all points lying in the interval. The query intervals (or aligned rectangles, as they are often called) may have arbitrary size, aspect ratio, and position. An obvious possibility for a multidimensional data structure is to map multidimensional points to one dimension, and to maintain the resulting one-dimensional points in one of the well-known data structures (such as, for instance, B+-trees). The mapping should be such that the query corresponding to the image of a multidimensional range in the given, original space can be supported by a data structure in image space. The ideal mapping (that does not exist) would map each multidimensional interval into a one-dimensional interval containing precisely the images of the corresponding points. With the goal of approximating this ideal in some way, several mappings have been proposed. The most

Space Filling Curves and Their Use in the Design of Geometric Data Structures

We are given a two-dimensional square grid of size N×N, where N∶=2n and n≥0. A space filling curve (SFC) is a numbering of the cells of this grid with numbers from c+1 to c+N2, for some c≥0. We call a SFC recursive (RSFC) if it can be recursively divided into four square RSFCs of equal size. Examples of well-known RSFCs include the Hilbert curve, the z-curve, and the Gray code.

Surface Modelling with Guaranteed Consistency - An Object-Based Approach

There have been many interpolation methods developed over the years, each with their own problems. One of the biggest limitations in many applications is the non-correspondence of the surface with the objects used to support it — usually a set of arbitrarily distributed data points. This is due to the metric methods used to define both the zone of influence of a data point and the set of data points used to estimate surface properties at intermediate locations. Most methods are coordinate system oriented, not object oriented.

On optimal cuts of hyperrectangles

We are given a set ofn d-dimensional (possibly intersecting) isothetic hyperrectangles. The topic of this paper is the separation of these rectangles by means of a cutting isothetic hyperplane. Thereby we assume that a rectangle which is intersected by the cutting plane iscut into two non-overlapping hyperrectangles. We investigate the behavior of several kinds of balancing functions, as well as their linear combination and present optimal and practical algorithms for computing the corresponding balanced cuts. In addition, we give tight worst-case bounds for the quality of the balanced cuts.ZusammenfassungGegeben sei eine Menge vonn (ggf. überlappenden) isothetischen Hyperrechtecken imd-dimensionalen Raum. Diese Arbeit beschäftigt sich mit Zerlegungen dieser Hyperrechteckmenge durch Schnitthyperebenen, wobei wir annehmen, daß jedes von einer Hyperebene geschnittene Hyperrechteck in zwei nicht-überlappende Hyperrechtecke zerschnitten wird. Wir untersuchen das Verhalten einiger Balancierungskriterien für Schnitte und präsentieren optimale and praktikable Algorithmen zur Berechnung der entsprechenden balancierten Schnitte. Schließlich geben wir auch scharfe Worst-case-Schranken für die bestmöglich erreichbare Qualität der balancierten Schnitte an.