OBDD-Based Representation of Interval Graphs

A graph $G = (V,E)$ can be described by the characteristic function of the edge set $\chi_E$ which maps a pair of binary encoded nodes to 1 iff the nodes are adjacent. Using \emph{Ordered Binary Decision Diagrams} (OBDDs) to store $\chi_E$ can lead to a (hopefully) compact representation. Given the OBDD as an input, symbolic/implicit OBDD-based graph algorithms can solve optimization problems by mainly using functional operations, e.g. quantification or binary synthesis. While the OBDD representation size can not be small in general, it can be provable small for special graph classes and then also lead to fast algorithms. In this paper, we show that the OBDD size of unit interval graphs is $O(\ | V \ | /\log \ | V \ |)$ and the OBDD size of interval graphs is $O(\ | V \ | \log \ | V \ |)$which both improve a known result from Nunkesser and Woelfel (2009). Furthermore, we can show that using our variable order and node labeling for interval graphs the worst-case OBDD size is$\Omega(\ | V \ | \log \ | V \ |)$. We use the structure of the adjacency matrices to prove these bounds. This method may be of independent interest and can be applied to other graph classes. We also develop a maximum matching algorithm on unit interval graphs using$O(\log \ | V \ |)$operations and a coloring algorithm for unit and general intervals graphs using$O(\log^2 \ | V \ |)$ operations and evaluate the algorithms empirically.Comment: 29 pages, accepted for 39th International Workshop on Graph-Theoretic Concepts 201

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

A General Introduction To Graph Visualization Techniques

Generally, a graph is an abstract data type used to represent relations among a given set of data entities. Graphs are used in numerous applications within the field of information visualization, such as VLSI (circuit schematics), state-transition diagrams, and social networks. The size and complexity of graphs easily reach dimensions at which the task of exploring and navigating gets crucial. Moreover, additional requirements have to be met in order to provide proper visualizations. In this context, many techniques already have been introduced. This survey aims to provide an introduction on graph visualization techniques helping the reader to gain a first insight into the most fundamental techniques. Furthermore, a brief introduction about navigation and interaction tools is provided

An overview of decision table literature.

The present report contains an overview of the literature on decision tables since its origin. The goal is to analyze the dissemination of decision tables in different areas of knowledge, countries and languages, especially showing these that present the most interest on decision table use. In the first part a description of the scope of the overview is given. Next, the classification results by topic are explained. An abstract and some keywords are included for each reference, normally provided by the authors. In some cases own comments are added. The purpose of these comments is to show where, how and why decision tables are used. Other examined topics are the theoretical or practical feature of each document, as well as its origin country and language. Finally, the main body of the paper consists of the ordered list of publications with abstract, classification and comments.

XML Matchers: approaches and challenges

Schema Matching, i.e. the process of discovering semantic correspondences between concepts adopted in different data source schemas, has been a key topic in Database and Artificial Intelligence research areas for many years. In the past, it was largely investigated especially for classical database models (e.g., E/R schemas, relational databases, etc.). However, in the latest years, the widespread adoption of XML in the most disparate application fields pushed a growing number of researchers to design XML-specific Schema Matching approaches, called XML Matchers, aiming at finding semantic matchings between concepts defined in DTDs and XSDs. XML Matchers do not just take well-known techniques originally designed for other data models and apply them on DTDs/XSDs, but they exploit specific XML features (e.g., the hierarchical structure of a DTD/XSD) to improve the performance of the Schema Matching process. The design of XML Matchers is currently a well-established research area. The main goal of this paper is to provide a detailed description and classification of XML Matchers. We first describe to what extent the specificities of DTDs/XSDs impact on the Schema Matching task. Then we introduce a template, called XML Matcher Template, that describes the main components of an XML Matcher, their role and behavior. We illustrate how each of these components has been implemented in some popular XML Matchers. We consider our XML Matcher Template as the baseline for objectively comparing approaches that, at first glance, might appear as unrelated. The introduction of this template can be useful in the design of future XML Matchers. Finally, we analyze commercial tools implementing XML Matchers and introduce two challenging issues strictly related to this topic, namely XML source clustering and uncertainty management in XML Matchers.Comment: 34 pages, 8 tables, 7 figure