Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik
Refine
Year of publication
Document Type
- Doctoral Thesis (57)
- Preprint (11)
- Article (1)
- Conference Proceeding (1)
Has Fulltext
- yes (70)
Keywords
- Maßtheorie (12)
- Graphenzeichnen (8)
- Iteriertes Funktionensystem (6)
- Quantisierung (6)
- Graph (5)
- Modellierung (5)
- Programmanalyse (5)
- Fraktale Dimension (4)
- Hausdorff-Dimension (4)
- Hausdorff-Maß (4)
In this thesis, we shall consider a certain class of algebraic cryptosystems called Gröbner Basis Cryptosystems. In 1994, Koblitz introduced the Polly Cracker cryptosystem that is based on the theory of Gröbner basis in commutative polynomials rings. The security of this cryptosystem relies on the fact that the computation of Gröbner basis is, in general, EXPSPACE-hard. Cryptanalysis of these commutative Polly Cracker type cryptosystems is possible by using attacks that do not require the computation of Gröbner basis for breaking the system, for example, the attacks based on linear algebra. To secure these (commutative) Gröbner basis cryptosystems against various attacks, among others, Ackermann and Kreuzer introduced a general class of Gröbner Basis Cryptosystems that are based on the difficulty of computing module Gröbner bases over general non-commutative rings. The objective of this research is to describe a special class of such cryptosystems by introducing the Weyl Gröbner Basis Cryptosystems. We divide this class of cryptosystems in two parts namely the (left) Weyl Gröbner Basis Cryptosystems and Two-Sided Weyl Gröbner Basis Cryptosystems. We suggest to use Gröbner bases for left and two-sided ideals in Weyl algebras to construct specific instances of such cryptosystems. We analyse the resistance of these cryptosystems to the standard attacks and provide computational evidence that secure Weyl Gröbner Basis Cryptosystems can be built using left (resp. two-sided) Gröbner bases in Weyl algebras.
In this thesis, we investigates plane drawings of undirected and directed graphs on cylinder surfaces. In the case of undirected graphs, the vertices are positioned on a line that is parallel to the cylinder’s axis and the edge curves must not intersect this line. We show that a plane drawing is possible if and only if the graph is a double-ended queue (deque) graph, i. e., the vertices of the graph can be processed according to a linear order and the edges correspond to items in the deque inserted and removed at their end vertices. A surprising consequence resulting from these observations is that the deque characterizes planar graphs with a Hamiltonian path. This result extends the known characterization of planar graphs with a Hamiltonian cycle by two stacks. By these insights, we also obtain a new characterization of queue graphs and their duals. We also consider the complexity of deciding whether a graph is a deque graph and prove that it is NP-complete. By introducing a split operation, we obtain the splittable deque and show that it characterizes planarity. For the proof, we devise an algorithm that uses the splittable deque to test whether a rotation system is planar. In the case of directed graphs, we study upward plane drawings where the edge curves follow the direction of the cylinder’s axis (standing upward planarity; SUP) or they wind around the axis (rolling upward planarity; RUP). We characterize RUP graphs by means of their duals and show that RUP and SUP swap their roles when considering a graph and its dual. There is a physical interpretation underlying this characterization: A SUP graph is to its RUP dual graph as electric current passing through a conductor to the magnetic field surrounding the conductor. Whereas testing whether a graph is RUP is NP-hard in general [Bra14], for directed graphs without sources and sink, we develop a linear-time recognition algorithm that is based on our dual graph characterization of RUP graphs.
Die Arbeit beschäftigt sich mit dem automatischen Zeichnen von Graphen. Hier wird ein interaktiver Ansatz untersucht, bei dem der Graph mit einer Menge von Operationen Schritt für Schritt aufgebaut wird. Der Zielgraph und die Einfügereihenfolge sind dabei nicht fest vorgegeben, sondern werden vom Benutzer bestimmt. In der Arbeit wird vor allem ein Szenario für zweifach zusammenhängende Graphen untersucht und ein für diese Zwecke passendes Zeichenmodell entwickelt. Dieser Ansatz wird dann um verschiedene Varianten erweitert. Außerdem wird gezeigt, dass das flächenminimale Zeichnen in dem neu entwickelten Zeichenmodell NP-vollständig ist.
In this dissertation we generalise the notion of level planar graphs in two directions: track planarity and radial planarity. Our main results are linear time algorithms both for the planarity test and for the computation of an embedding, and thus a drawing. Our algorithms use and generalise PQ-trees, which are a data structure for efficient planarity tests.
The increasing cost of energy and the worldwide desire to reduce CO2 emissions has raised concern about the energy efficiency of information and communication technology. Whilst research has focused on data centres recently, this thesis identifies office computing environments as significant consumers of energy. Office computing environments offer great potential for energy savings: On one hand, such environments consist of a large number of hosts. On the other hand, these hosts often remain turned on 24~hours per day while being underutilised or even idle. This thesis analyzes the energy consumption within office computing environments and suggests an energy-efficient virtualized office environment. The office environment is virtualized to achieve flexible virtualized office resources that enable an energy-based resource management. This resource management stops idle services and idle hosts from consuming resources within the office and consolidates utilised office services on office hosts. This increases the utilisation of some hosts while other hosts are turned off to save energy. The suggested architecture is based on a decentralized approach that can be applied to all kinds of office computing environments, even if no centralized data centre infrastructure is available. The thesis develops the architecture of the virtualized office environment together with an energy consumption model that is able to estimate the energy consumption of hosts and network within office environments. The model enables the energy-related comparison of ordinary and virtualized office environments, considering the energy-efficient management of services. Furthermore, this thesis evaluates energy efficiency and overhead of the suggested approach. First, it theoretically proves the energy efficiency of the virtualized office environment with respect to the energy consumption model. Second, it uses Markov processes to evaluate the impact of user behaviour on the suggested architecture. Finally, the thesis develops a discrete-event simulation that enables the simulation and evaluation of office computing environments with respect to varying virtualization approaches, resource management parameters, user behaviour, and office equipment. The evaluation shows that the virtualized office environment saves more than half of the energy consumption within office computing environments, depending on user behaviour and office equipment.
This work presents techniques for the construction of a global data integrations system. Similar to distributed databases this system allows declarative queries in order to express user-specific information needs. Scalability towards global data integration systems and openness were major design goals for the architecture and techniques developed in this work. It is shown how service composition, extensibility and quality of service can be supported in an open system of providers for data, functionality for query processing operations, and computing power.
The Sugiyama framework proposed in the seminal paper of 1981 is one of the most important algorithms in graph drawing and is widely used for visualizing directed graphs. In its common version, it draws graphs hierarchically and, hence, maps the topological direction to a geometric direction. However, such a hierarchical layout is not possible if the graph contains cycles, which have to be destroyed in a preceding step. In certain application and problem settings, e.g., bio sciences or periodic scheduling problems, it is important that the cyclic structure of the input graph is preserved and clearly visible in drawings. Sugiyama et al. also suggested apart from the nowadays standard horizontal algorithm a cyclic version they called recurrent hierarchies. However, this cyclic drawing style has not received much attention since. In this thesis we consider such cyclic drawings and investigate the Sugiyama framework for this new scenario. As our goal is to visualize cycles directly, the first phase of the Sugiyama framework, which is concerned with removing such cycles, can be neglected. The cyclic structure of the graph leads to new problems in the remaining phases, however, for which solutions are proposed in this thesis. The aim is a complete adaption of the Sugiyama framework for cyclic drawings. To complement our adaption of the Sugiyama framework, we also treat the problem of cyclic level planarity and present a linear time cyclic level planarity testing and embedding algorithm for strongly connected graphs.
Die vorliegende Arbeit erschließt durch die Formalisierung einer linguistischen Theorie Möglichkeiten zum Entwurf generischer Verfahren zur Verarbeitung natürlicher Sprachen. Zu diesem Zweck setzen wir Graphsprachen für die Modellierung syntaktischer Strukturen ein. Damit lassen sich Ergebnisse der linguistischen Forschung mit Begriffen der Graphentheorie beschreiben und bewerten. Zu diesem Ansatz motiviert der Umstand, daß in der Linguistik im Rahmen der Syntax jedem Satz einer natürlichen Sprache eine nichtsequentielle Struktur zugesprochen wird. Diese Struktur überlagert die lineare Wortfolge, die wir als Satz kennen. Eine Menge solcher syntaktischen Strukturen die wir mit Graphen modellieren können betrachten wir als Graphsprache. Die Arbeit zeigt, wie sich solche Graphsprachen mit Hilfe von Graphgrammatiken beschreiben lassen. Wie alle formalen Sprachen zeichnen sich auch Graphgrammatiken dadurch aus, daß sie mathematisch wohldefniert sind. Dies stellt eine notwendige Voraussetzung dar, um Aussagen über eine Sprache zu beweisen. Von Interesse ist dabei vor allem die Untersuchung unendlicher Mengen. Das Ziel besteht dann darin, für sie eine endliche Beschreibung zu finden. Diese Aufgabe wird in der Regel von einer Grammatik erfüllt. Darüber hinaus ist man an erkennenden Algorithmen für Sprachen interessiert, die das Wortproblem effizient lösen. Bezüglich natürlicher Sprachen werden beide Aufgabenstellungen in dieser Arbeit mit Hilfe von Graphgrammatiken gelöst.
Real Closed * Rings
(2010)
In this dissertation I examine a definition of real closure of commutative unitary reduced rings. I also give a characterization of rings that are real closed in this context and how one is able to arrive to such a real closure. There are sufficient examples to help the reader get a feel for real closed * rings and the real closure * of commutative unitary rings.