Geometric Secluded Paths and Planar Satisfiability

We consider paths with low exposure to a 2D polygonal domain, i.e., paths which are seen as little as possible; we differentiate between integral exposure (when we care about how long the path sees every point of the domain) and 0/1 exposure (just counting whether a point is seen by the path or not). For the integral exposure, we give a PTAS for finding the minimum-exposure path between two given points in the domain; for the 0/1 version, we prove that in a simple polygon the shortest path has the minimum exposure, while in domains with holes the problem becomes NP-hard. We also highlight connections of the problem to minimum satisfiability and settle hardness of variants of planar min- and max-SAT

Visibility Domains and Complexity

Two problems in discrete and computational geometry are considered that are related to questions about the combinatorial complexity of arrangements of visibility domains and about the hardness of path planning under cost measures defined using visibility domains. The first problem is to estimate the VC-dimension of visibility domains. The VC-dimension is a fundamental parameter of every range space that is typically used to derive upper bounds on the size of hitting sets. Better bounds on the VC-dimension directly translate into better bounds on the size of hitting sets. Estimating the VC-dimension of visibility domains has proven to be a hard problem. In this thesis, new tools to tackle this problem are developed. Encircling arguments are combined with decomposition techniques of a new kind. The main ingredient of the novel approach is the idea of relativization that makes it possible to replace in the analysis of intersections the complicated visibility domains by simpler geometric ranges. The main result here is the new upper bound of 14 on the VC-dimension of visibility polygons in simple polygons that improves significantly upon the previously known best upper bound of 23. For the VC-dimension of perimeter visibility domains, the new techniques yield an upper bound of 7 that leaves only a very small gap to the best known lower bound of 5. The second problem considered is to compute the barrier resilience of visibility domains. In barrier resilience problems, one is given a set of barriers and two points s and t in R^d. The task is to find the minimum number of barriers one has to remove such that there is a way between s and t that does not cross a barrier. In the field of sensor networks, the barriers are interpreted as sensor ranges and the barrier resilience of a network is a measure for its vulnerability. In this thesis the very natural special case where the barriers are visibility domains is investigated. It can also be formulated in terms of finding a so-called minimum witness path. For visibility domains in simple polygons it is shown that one can find an optimal path efficiently. For polygons with holes an approximation hardness result is shown that is stronger than previous hardness results in geometric settings. Two different three-dimensional settings are considered and their respective relations to the Minimum Neighborhood Path problem and the Minimum Color Path problem in graphs are demonstrated. For one of the three-dimensional problems a 2-approximation algorithm is designed. For the general problem of finding minimum witness paths among polyhedral obstacles it turns out that it is not approximable in a strong sense

The Limited Workspace Model for Geometric Algorithms

Space usage has been a concern since the very early days of algorithm design. The increased availability of devices with limited memory or power supply – such as smartphones, drones, or small sensors – as well as the proliferation of new storage media for which write access is comparatively slow and may have negative effects on the lifetime – such as flash drives – have led to renewed interest in the subject. As a result, the design of algorithms for the limited workspace model has seen a significant rise in popularity in computational geometry over the last decade. In this setting, we typically have a large amount of data that needs to be processed. Although we may access the data in any way and as often as we like, write-access to the main storage is limited and/or slow. Thus, we opt to use only higher level memory for intermediate data (e.g., CPU registers). Since the application areas of the devices mentioned above – sensors, smartphones, and drones – often handle a large amount of geographic (i.e., geometric) data, the scenario becomes particularly interesting from the viewpoint of computational geometry. Motivated by these considerations, we investigate geometric problems in the limited workspace model. In this model the input of size n resides in read-only memory, an algorithm may use a workspace of size s = {1, . . . , n} to read and write the intermediate data during its execution, and it reports the output to a write-only stream. The goal is to design algorithms whose running time decreases as s increases, which provides a time-space trade-off. In this thesis, we consider three fundamental geometric problems, namely, computing different types of Voronoi diagrams of a planar point set, computing the Euclidean minimum spanning tree of a planar point set, and computing the k-visibility region of a point inside a polygonal domain. Using several innovative techniques, we either achieve the first time-space trade-offs for those problems or improve the previous results.Der Speicherplatzbedarf ist seit den Anfängen des Algorithmenentwurfs von Interesse. Die erhöhte Verfügbarkeit von Geräten mit begrenztem Speicherplatz oder begrenzter Stromversorgung – wie Smartphones, Drohnen oder kleine Sensoren – sowie die Verbreitung neuer Speichermedien, bei denen der Schreibzugriff vergleichsweise langsam ist und negative Auswirkungen auf die Lebensdauer haben kann – wie beispielsweise Flash-Laufwerken – haben zu erneuter Aufmerksamkeit für dieses Thema geführt. In der Folge hat der Entwurf von Algorithmen für das Limited Workspace Model (Modell mit begrenztem Arbeitsspeicher) in den letzten zehn Jahren einen signifikanten Anstieg an Popularität in der algorithmischen Geometrie erfahren. In diesem Setting haben wir in der Regel eine große Menge an Daten, die verarbeitet werden müssen. Obwohl wir auf die Daten beliebig oft und in beliebiger Weise zugreifen können, ist der Schreibzugriff auf den Hauptspeicher begrenzt und/oder langsam. Zwischenergebnisse werden daher nur in einem kleineren, übergeordneten Speicher (z. B. CPU-Register) abgelegt. Da die Anwendungsbereiche der oben genannten Geräte – Sensoren, Smartphones und Drohnen – oft mit einer großen Menge an geografischen (d. h., geometrischen) Daten umgehen, ist dieses Szenario aus Sicht der algorithmischen Geometrie besonders interessant. Motiviert durch diese Überlegungen haben wir geometrische Probleme im Limited Workspace Model untersucht. In diesem Modell befindet sich die Eingabe der Größe n in einem schreibgeschützten Speicher, ein Algorithmus kann einen Arbeitsspeicher der Größe s = {1, . . . , n} verwenden, um die Zwischendaten während der Ausführung zu lesen und zu schreiben. Die Ausgabe sendet er an einen lesegeschützten Stream. Ziel ist es, Algorithmen zu entwickeln, deren Laufzeit mit zunehmender Verfügbarkeit an Arbeitsspeicher abnimmt, was einen Time-Space Trade-Off (Laufzeit-Speicher-Abwägung) darstellt. In dieser Arbeit betrachten wir drei grundlegende geometrische Probleme, nämlich die Berechnung verschiedener Arten von Voronoi-Diagrammen einer Punktmenge in der Ebene, die Berechnung des euklidischen minimalen Spannbaums einer ebenen Punktmenge und die Bestimmung der k-Sichtbarkeitsregion (k-visibility region) eines Punkts innerhalb eines polygonalen Gebiets. Mit mehreren innovativen Techniken entwickeln wir entweder die ersten Time-Space Trade-Offs für diese Probleme oder verbessern die bisherigen Ergebnisse

Higher-order Voronoi diagrams of polygonal objects

Higher-order Voronoi diagrams are fundamental geometric structures which encode the k-nearest neighbor information. Thus, they aid in computations that require proximity information beyond the nearest neighbor. They are related to various favorite structures in computational geometry and are a fascinating combinatorial problem to study. While higher-order Voronoi diagrams of points have been studied a lot, they have not been considered for other types of sites. Points lack dimensionality which makes them unable to represent various real-life instances. Points are the simplest kind of geometric object and therefore higher- order Voronoi diagrams of points can be considered as the corner case of all higher-order Voronoi diagrams. The goal of this dissertation is to move away from the corner and bring the higher-order Voronoi diagram to more general geometric instances. We focus on certain polygonal objects as they provide flexibility and are able to represent real-life instances. Before this dissertation, higher-order Voronoi diagrams of polygonal objects had been studied only for the nearest neighbor and farthest Voronoi diagrams. In this dissertation we investigate structural and combinatorial properties and discover that the dimensionality of geometric objects manifests itself in numerous ways which do not exist in the case of points. We prove that the structural complexity of the order-k Voronoi diagram of non-crossing line segments is O(k(n-k)), as in the case of points. We study disjoint line segments, intersecting line segments, line segments forming a planar straight-line graph and extend the results to the Lp metric, 1<=p<=infty. We also establish the connection between two mathematical abstractions: abstract Voronoi diagrams and the Clarkson-Shor framework. We design several construction algorithms that cover the case of non-point sites. While computational geometry provides several approaches to study the structural complexity that give tight realizable bounds, developing an effective construction algorithm is still a challenging problem even for points. Most of the construction algorithms are designed to work with points as they utilize their simplicity and relations with data-structures that work specifically for points. We extend the iterative and the sweepline approaches that are quite efficient in constructing all order-i Voronoi diagrams, for i<=k and we also give three randomized construction algorithms for abstract higher-order Voronoi diagrams that deal specifically with the construction of the order-k Voronoi diagrams