Marc J. van Kreveld

Contour trees and small seed sets for isosurface traversal

For 2D or 3D meshes that represent the domain of continuous function to the reals, the contours|or isosurfaces|of a speci ed value are an important way to visualize the function. To nd such contours, a seed set can be used for the starting points from which the traversal of the contours can begin. This paper gives the rst methods to obtain seed sets that are provably small in size. They are based on a variant of the contour tree (or topographic change tree). We give a new, simple algorithm to compute such a tree in regular and irregular meshes that requires O(n logn) time in 2D for meshes with n elements, and in O(n) time in higher dimensions. The additional storage overhead is proportial to the maximum size of any contour (linear in the worst case, but typically less). Given the contour tree, a minimum size seed set can be computed in roughly quadratic time. Since in practice this can be excessive, we develop a simple approximation algorithm giving a seed set of size at most twice the size of the minimum. It requires O(n log n) time and linear storage once the contour tree is known. We also give experimental results, showing the size of the seed sets for several data sets.

Computing longest duration flocks in trajectory data

Moving point object data can be analyzed through the discovery of patterns. We consider the computational efficiency of computing two of the most basic spatio-temporal patterns in trajectories, namely flocks and meetings. The patterns are large enough subgroups of the moving point objects that exhibit similar movement and proximity for a certain amount of time. We consider the problem of computing a longest duration flock or meeting. We give several exact and approximation algorithms, and also show that some variants are as hard as MaxClique to compute and approximate.

Label placement by maximum independent set in rectangles

Motivated by the problem of labeling maps, we investigate the problem of computing a large non-intersecting subset in a set of n rectangles in the plane. Our results are as follows. In O(n log n) time, we can find an O(log n)factor approximation of the maximum subset in a set of n arbitrary axis-parallel rectangles in the plane. If all rectangles have unit height, we can find a 2-approximation in O(n logn) time. Extending this result, we obtain a (1 + l/k)-approximati on in time O(n logn + n 2k-1) time, for any integer k/> 1.

Finding REMO — Detecting Relative Motion Patterns in Geospatial Lifelines

Technological advances in position aware devices increase the availability of tracking data of everyday objects such as animals, vehicles, people or football players. We propose a geographic data mining approach to detect generic aggregation patterns such as flocking behaviour and convergence in geospatial lifeline data. Our approach considers the object’s motion properties in an analytical space as well as spatial constraints of the object’s lifelines in geographic space. We discuss the geometric properties of the formalised patterns with respect to their efficient computation.

Efficient detection of motion patterns in spatio-temporal data sets

Moving point object data can be analyzed through the discovery of patterns. We consider the computational efficiency of detecting four such spatio-temporal patterns, namely flock, leadership, convergence, and encounter, as defined by Laube et al., 2004. These patterns are large enough subgroups of the moving point objects that exhibit similar movement in the sense of direction, heading for the same location, and/or proximity. By the use of techniques from computational geometry, including approximation algorithms, we improve the running time bounds of existing algorithms to detect these patterns.

Point labeling with sliding labels

This paper discusses algorithms for labeling sets of points in the plane, where labels are not restricted to some nite number of positions. We show that continuously sliding labels allow more points to be labeled both in theory and in practice. We dene six dierent models of labeling. We compare models by analyzing how many more points can receive labels under one model than another. We show that maximizing the number of labeled points is NP-hard in the most general of the new models. Nevertheless, we give a polynomial-time approximation scheme and a simple and ecient factor- 1 approximation algorithm for each of the new models. Finally, we give experimental results based on the factor- 1 approximation algorithm to compare the models in practice. We also compare this algorithm experimentally to other algorithms suggested in the literature.

Efficient Detection of Patterns in 2D Trajectories of Moving Points

Moving point object data can be analyzed through the discovery of patterns in trajectories. We consider the computational efficiency of detecting four such spatio-temporal patterns, namely flock, leadership, convergence, and encounter, as defined by Laube et al., Finding REMO—detecting relative motion patterns in geospatial lifelines, 201–214, (2004). These patterns are large enough subgroups of the moving point objects that exhibit similar movement in the sense of direction, heading for the same location, and/or proximity. By the use of techniques from computational geometry, including approximation algorithms, we improve the running time bounds of existing algorithms to detect these patterns.

On rectangular cartograms

A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable (e.g., population). Good rectangular cartograms are hard to generate: The area specifications for each rectangle may make it impossible to realize correct adjacencies between the regions and so hamper the intuitive understanding of the map. We present the first algorithms for rectangular cartogram construction. Our algorithms depend on a precise formalization of region adjacencies and build upon existing VLSI layout algorithms. Furthermore, we characterize a non-trivial class of rectangular subdivisions for which exact cartograms can be computed efficiently. An implementation of our algorithms and various tests show that in practice, visually pleasing rectangular cartograms with small cartographic error can be generated effectively.

Largest and Smallest Convex Hulls for Imprecise Points

Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(nlog n) to O(n13), and prove NP-hardness for some other variants.

Web-based Delineation of Imprecise Regions

This paper describes several steps in the derivation of boundaries of imprecise regions using the Web as the information source. We discuss how to use the Web to obtain locations that are part of and locations that are not part of the region to be delineated, and then we propose methods to compute the region algorithmically. The methods introduced are evaluated to judge the potential of the approach.