Delaunay triangulation of imprecise points in linear time after preprocessing

An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of disjoint unit disks in the plane in time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay

Complexity and Algorithms for the Discrete Fr\'echet Distance Upper Bound with Imprecise Input

We study the problem of computing the upper bound of the discrete Fr\'{e}chet distance for imprecise input, and prove that the problem is NP-hard. This solves an open problem posed in 2010 by Ahn \emph{et al}. If shortcuts are allowed, we show that the upper bound of the discrete Fr\'{e}chet distance with shortcuts for imprecise input can be computed in polynomial time and we present several efficient algorithms.Comment: 15 pages, 8 figure

Recent progress in exact geometric computation

AbstractComputational geometry has produced an impressive wealth of efficient algorithms. The robust implementation of these algorithms remains a major issue. Among the many proposed approaches for solving numerical non-robustness, Exact Geometric Computation (EGC) has emerged as one of the most successful. This survey describes recent progress in EGC research in three key areas: constructive zero bounds, approximate expression evaluation and numerical filters

Multi-scale persistent homology

Data has shape and that shape is important. This is the anthem of Topological Data Analysis (TDA) as often stated by Gunnar Carlsson. In this paper we take a common method of persistence involving the growing of balls of the same size, and generalizing to balls of different sizes in order to better understand the outlier and coverage problems. We begin with a summary of classical persistence theory and stability. We then move on to generalizing the Rips and \v{C}ech complexes as well as generalizing the Rips lemma. We transition into 3 notions of stability in terms of bottleneck distance. For the outlier problem, we show that it is possible to interpolate between persistence on a set with no noise and a set with noise. For the coverage problem, we present an algorithm which provides a cheap way of covering a compact domain

Distributed multi-UAV shield formation based on virtual surface constraints

This paper proposes a method for the deployment of a multi-agent system of unmanned aerial vehicles (UAVs) as a shield with potential applications in the protection of infrastructures. For this purpose, a distributed control law based on the gradient of a potential function is proposed to acquire the desired shield shape, which is modeled as a quadric surface in the 3D space. The graph of the formation is a Delaunay triangulation, which guarantees the formation to be rigid. An algorithm is proposed to design the formation (target distances between agents and interconnections) to distribute the agents over the virtual surface, where the input parameters are just the parametrization of the quadric and the number of agents of the system. Proofs of system stability with the proposed control law are provided, as well as a new method to guarantee that the resulting triangulation over the surface is Delaunay, which can be executed locally. Simulation and experimental results illustrate the effectiveness of the proposed approach

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

Jump flooding algorithm on graphics hardware and its applications

Ph.DDOCTOR OF PHILOSOPH