The amount of data that are being generated, collected, and stored is growing rapidly and a large body of the data are geometric in nature. In many applications, in particular those involving geographic data, the data needs to be stored in such a way that relevant parts of the data can be retrieved quickly. More precisely, one often wants to store the data in such a way that certain queries about the data can be answered quickly. Thus the art of designing efficient geometric data structures has always been a core topic within computational geometry. Unfortunately, traditional data structures are far from being able to get involved in many modern applications, as they are neither able to store complex objects nor to support queries that involve complicated analysis tasks. Inspired by the two downsides of traditional data structures, in this thesis we design and analyze a number of new efficient data structures that go beyond traditional setting.
A schematic map focuses on displaying specific information rather than full geographic detail. Such a map uses few geometric elements to convey a geographic shape. The geometry often adheres to a certain style, such as the octilinear style which only permits horizontal, vertical and 45-degree diagonal lines. For a map to be effective, geographic shapes such as countries and provinces must still be recognizable. In this thesis we investigate similarity measures and algorithms in the context of automated cartographic schematization, the process of creating schematic maps. Similarity measures are required to quantify resemblance between schematic and geographic shapes. We improve an existing similarity measure and develop new algorithms to compute schematic representations that resemble the geographic shapes.
Over the course of the years many different structures have been developed for computing and storing geometric information on a set of objects (often points). In recent years there has been a growing interest in maintaining such structures as the input moves over time. The straightforward approach is to recompute the desired structure from scratch whenever the input changes, but this can be wasteful if the changes are small. To avoid unnecessary computations the Kinetic Data Structures (KDS) framework was introduced by Basch, Guibas and Hershberger in 1997. In this framework we assume that the trajectories of the objects are known and we can compute when to change the data structure. This leads to an event-driven approach for which several efficiency guarantees can be proven. Unfortunately, there are many situations when the trajectories are unknown. For example when the input comes from tracking devices, human-controlled avatars in a virtual world or a numerical integrator in a physical simulation. In these cases new locations become available as time progresses and no precise trajectories can be given beforehand. This means we cannot apply the KDS-framework, but we would still like to maintain data structures in a provably efficient manner. For these applications we introduce the black-box model to model the motions of the input and design efficient algorithms to maintain several basic data structures within this model.
This thesis tackles two problems that seem unrelated at the surface. The first is from robotics: a disk-shaped robot needs to move a disk-shaped object, too heavy for it to pick up. To this end, we compute a path that will make it push the object to the desired location, avoiding obstacles along the way. The second problem is related to air-traffic control and interactive maps. A set of points on the screen, representing air planes or cities, is to be assigned textual labels. As the air planes move, or the user's view onto the cities is panned, rotated, or zoomed, the labels need to be moved smoothly, as if pulled by the points. By exploiting the symmetry between pushing and pulling, we can re-use some of the algorithmic techniques used to solve the one problem in solving the other.
Maps play an important role in our society. We use maps to find our way from one place to another, or to locate points of interest, like hotels, restaurants, and tourist attractions. But maps not only help us navigate through our environment. They are effective tools for communicating various types of information and help people to make decisions in, for example, spatial planning and politics. Recent technological advances and the changing needs of map users require maps to be created on-demand. Hence we need methods for the automated construction of maps. In this thesis we consider the automated construction of thematic maps, in particular schematic maps, cartograms, necklace maps, and flow maps. We study the underlying algorithmic problems and we present efficient algorithms to construct these maps.
Spatial data structures form a core ingredient of many geometric algorithms, both in theory and in practice. Many of these data structures, especially the ones used in practice are based on partitioning the underlying space (examples are binary space partitions and decompositions of polygons) or partitioning the set of objects (examples are bounding-volume hierarchies). The efficiency of such data structures—and, hence, of the algorithms that use them—depends on certain characteristics of them such as their size or their crossing number. Much research has been done on the problem of finding such data structures whose characteristics are good in the worst case. In this thesis, we studied the problem from a different point of view, namely instance-optimality. In particular, we considered the following question: given a class of geometric partitioning structures—like BSPs, simplicial partitions, polygon decompositions, …—and a cost function—like size or crossing number—can we design an algorithm that computes a structure whose cost is optimal or close to optimal for any input instance (rather than only worst-case optimal).
Landscapes and their morphology have been widely studied for predicting physical phenomena, such as floods or erosion, but also for planning human activities effectively, such as building prominent fortifications and watchtowers. Nowadays, the study of terrains is done in a computer-based environment; terrains are modelled by digital representations, and algorithms are used to simulate physical processes like water flow and to compute attributes like visibility from certain locations. One of the most popular digital terrain representations are the so-called Triangulated Irregular Networks (TINs), that is, piecewise linear surfaces that consist of triangles. In the current thesis we examine several complexity issues related with the computation of flow and visibility structures on triangulated terrains. Among other results, we provide novel algorithms for extracting information on the drainage attributes of such terrains efficiently. We also study the influence of noise in the computation of drainage and visibility structures on triangulated terrains.
For many problems of practical interest it seems to be impossible to compute their optimal solution in polynomial time. A c-approximation algorithm is a polynomial-time algorithm that produces a solution which may not be optimal, but is never more than a factor c off from the optimum. The factor c is then called the approximation ratio. A randomized c-approximation algorithm is an approximation algorithm which makes decisions based on the outcome of random events (say, coin tosses) so that its approximation ratio is a random variable with expected value c. In this thesis, randomized approximation algorithms are given for a number of combinatorial optimization problems, including metric uncapacitated facility location, construction of phylogenetic networks, computation of Nash equilibria, and a generalization of set cover. The algorithms can be derandomized by the method of conditional expectation to give guaranteed approximation ratios that often improve on the previously best known.
Graphs are ubiquitous in computer science, and visualizing them in a good way has become the topic of the research area known as graph drawing. It combines techniques from graph theory, algorithms, (computational) geometry, and (computational) topology, and has applications to cartography, VLSI design, and information visualization, among others. This thesis studies three graph drawing problems. The first problem is the construction of rectilinear cartograms, where each region on a map is deformed into a rectilinear polygon whose area represents some value associated with the area (say, the population of a country). The resulting cartogram should resemble the original map as closely as possible in terms of shapes and adjacencies of the regions, while at the same time faithfully representing the value associated with each region. The second problem concerns cased drawings of graphs, where edge crossings are drawn in a way that suggests that one edge goes underneath the other. The third problem is to decide for certain curves and surfaces whether they could occur as the boundary of some shape.
Traditionally, the analysis of the running time of an algorithm is a worst-case analysis. Because of this, the bound on the running time is determined by the most difficult inputs, even when such inputs are non-existent in the application at hand. This is true in all areas of algorithms research, but it is particularly problematic for geometric algorithms. The reason is that there can be a large variation in shape and spatial distribution of geometric data, and unfavorable properties of the input—extremely skinny objects for example—can increase the running time dramatically. In this thesis we design and analyze several novel geometric algorithms under the assumption that the input objects are fat, that is, they do not contain extremely skinny parts. In particular, we show how to decompose fat objects into simpler pieces (such as triangles or tetrahedra), and we give algorithms for ray shooting, computing depth orders, and hidden-surface removal for fat objects. All our algorithms are significantly more efficient than the best known solutions for arbitrary (non-fat) objects.
A geometric network on a set P of n points in d-dimensional space is an undirected graph G=(P,E) with vertex set P whose edges are straight-line segments connecting pairs of points in P. Geometric networks can model many real-life networks: road networks, telecommunication networks, and so on. When designing a network for a given point set P, it is often important to ensure a fast connection between every pair of points. For this it would be ideal to have a direct connection between every pair—the network would then be a complete graph—but in most applications this is unacceptable due to the high costs. This leads to the concepts of spanners: networks with a small number of edges where there is a relatively short path between any pair of points. In this thesis we obtain several new results on geometric spanners. For example, we develop novel algorithms for constructing spanners that are fault-tolerant and for adding edges to an existing spanner to improve its performance, and we experimentally investigate the performance of different spanner constructions.
Motion is ubiquitous in the physical world and due to recent advances in sensing and tracking technology, motion data is becoming more and more available in a variety of areas: air-traffic control, mobile communication, geographic information systems, and so on. It is not surprising, therefore, that in these applications areas of computer science it is necessary to store, analyze, and create or manipulate motion data. As a result, algorithms and data structures for moving objects have become an important area of study within the field of computational geometry. This thesis presents a number of new and fundamental results in this exciting area of research.
Many computational problems can be stated in terms of geometric queries on a set of objects, such as: ‘Which objects in the set are contained within this region?’. This query is an example of a range searching query. If a set of objects is often queried then it is beneficial to build a data structure on these objects such that the queries can be answered faster. This can be a very specific data structure, that only answers one specific type of geometric query. However, it can also be a general data structure, that answers many different queries and stores many types of objects. The latter type of data structures we call multifunctional geometric data structures and these are the topic of this thesis