Drawing Trees with Perfect Angular Resolution and Polynomial Area

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2{\pi}/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure

Quad general tree drawing algorithm and general trees characterization: towards an environment for the experimental study on general tree drawing algorithms

Information visualization produces (interactive) visual representations of abstract data to reinforce human cognition and perception; thus enabling the viewer to gain knowledge about the internal structure of the data and causal relationships in it. The visualization of information hierarchies is concerned with the presentation of abstract hierarchical information about relationships between various entities. It has many applications in diverse domains such as software engineering, information systems, biology, and chemistry. Information hierarchies are typically modeled by an abstract tree, where vertices are entities and edges represent relationships between entities. The aim of visualizing tree drawings is to automatically produce drawings of trees which clearly reflect the relationships of the information hierarchy. This thesis is primarily concerned with introducing the new general tree drawing algorithm Quad that produces good visually distinguishable angles, and a characterization of general trees which allows us to classify general trees into several types based on their characteristics. Both of these topics are part of building an experimental study environment for the evaluation of drawing algorithms for general trees. The main achievements of this thesis include: 1. A study on characterization of general trees that aims to classify them into several types. 2. A tree drawing algorithm that produces visually distinguishable angles for high degree general trees with user specified angular coefficient

Interactive visualization of information hierarchies and applications on the web

The visualization of information hierarchies is concerned with the presentation of abstract hierarchical information about relationships between various entities. It has many applications in diverse domains such as software engineering, information systems, biology, and chemistry. Information hierarchies are typically modeled by an abstract tree, where vertices are entities and edges represent relationships between entities. The aim of visualizing tree drawings is to automatically produce drawings of trees which clearly reflect the relationships of the information hierarchy. This thesis is primarily concerned with problems related to the automatic generation of area-efficient grid drawings of trees, interactively visualizing information hierarchies, and applying our techniques on Web data. The main achievements of this thesis include: 1. An experimental study on algorithms that produce planar straight-line grid drawings of binary trees, 2. An experimental study that shows the algorithm for producing planar straight-line grid drawings of degree-d trees with n nodes with optimal linear area and with user-defined arbitrary aspect ratio, works well in practice, 3. A rings-based technique for interactively visualizing information hierarchies, in real-time, 4. A survey of Web visualization systems developed to address the lost in cyberspace problem, 5. A separation-based Web visualization system that we present as a viable solution to the lost in cyberspace problem, 6. A rings-based Web visualization system that we propose as a solution to the lost in cyberspace problem

A Potential-Field-Based Multilevel Algorithm for Drawing Large Graphs

The aim of automatic graph drawing is to compute a well-readable layout of a given graph G=(V,E). One very popular class of algorithms for drawing general graphs are force-directed methods. These methods generate drawings of G in the plane so that each edge is represented by a straight line connecting its two adjacent nodes. The computation of the drawings is based on associating G with a physical model. Then, the algorithms iteratively try to find a placement of the nodes so that the total energy of the physical system is minimal. Several force-directed methods can visualize large graphs containing many thousands of vertices in reasonable time. However, only some of these methods guarantee a sub-quadratic running time in special cases or under certain assumptions, but not in general. The others are not sub-quadratic at all. We develop a new force-directed algorithm that is based on a combination of an efficient multilevel strategy and a method for approximating the repulsive forces in the system by rapidly evaluating potential fields. The worst-case running time of the new method is O(|V| log|V|+|E|) with linear memory requirements. In practice, the algorithm generates nice drawings of graphs containing up to 100000 nodes in less than five minutes. Furthermore, it clearly visualizes even the structures of those graphs that turned out to be challenging for other tested methods

A Potential-Field-Based Multilevel Algorithm for Drawing Large Graphs

The aim of automatic graph drawing is to compute a well-readable layout of a given graph G=(V,E). One very popular class of algorithms for drawing general graphs are force-directed methods. These methods generate drawings of G in the plane so that each edge is represented by a straight line connecting its two adjacent nodes. The computation of the drawings is based on associating G with a physical model. Then, the algorithms iteratively try to find a placement of the nodes so that the total energy of the physical system is minimal. Several force-directed methods can visualize large graphs containing many thousands of vertices in reasonable time. However, only some of these methods guarantee a sub-quadratic running time in special cases or under certain assumptions, but not in general. The others are not sub-quadratic at all. We develop a new force-directed algorithm that is based on a combination of an efficient multilevel strategy and a method for approximating the repulsive forces in the system by rapidly evaluating potential fields. The worst-case running time of the new method is O(|V| log|V|+|E|) with linear memory requirements. In practice, the algorithm generates nice drawings of graphs containing up to 100000 nodes in less than five minutes. Furthermore, it clearly visualizes even the structures of those graphs that turned out to be challenging for other tested methods

Algorithms for visualization of graph-based structures

Buildings today are built to maintain a healthy indoor environment and an efficient energy usage which is probably why damages caused by dampness has increased since the 1960’s. A study between year 2008 and 2010 showed that 26 percent of the 110 000 examined houses had damages and flaws caused by dampness that could prove to be harmful later on. This means that one out of four bathrooms risk the chance to develop damages by dampness. Approximately 2 percent of the houses had already developed water damages. It is here where the problems appear. A house or a building that is damaged by water of dampness need time to dry out before any renovation can take place. This means that damaged parts must be removed and allowed to dry out, this takes a long time to do and the costs are high and at the same time it can cause inconvenience to the residents. Here is where the Air Gap Method enters the picture. The meaning with the method is to drain and dry out the moisture without the need to perform a larger renovation. The Air Gap Method is a so called "forgiving"-system that is if water damages occur the consequences will be small. The Air Gap method means that an air gap is created in the walls, ceiling and the floor where a heating cable in the gap heats up the air and creates an air movement. The point is to create a stack effect in the gap that with the help of the air movement transports the damp air through an opening by the ceiling. The aim of this thesis is to examine if it’s necessary with the heating cable in the air gap and if there is a specific drying out pattern of the water damaged bathroom floor. The possibility of mould growth will also be examined. The study showed that the damped floor did dry out even without a heating cable, but as one of the studies showed signs of mould growth it is shown that the risk for mould growth is higher without a heating cable. There was a seven days difference in the drying out time between the studies with and without the heating cable; this difference can be decisive for mould growth which is why the heating cable is recommended. The Air Gap method is quite easy to apply in houses with light frame constructions simply by using a smaller dimension on the studs to create the air gap in the floor and walls. The method can also be applied in apartment buildings with a concrete frame by using the room-in- room principal. When renovating existing bathrooms it’s easier to use prefabricated elements to create the air gap in the floor and walls. ~

Schematics of Graphs and Hypergraphs

Graphenzeichnen als ein Teilgebiet der Informatik befasst sich mit dem Ziel Graphen oder deren Verallgemeinerung Hypergraphen geometrisch zu realisieren. Beschränkt man sich dabei auf visuelles Hervorheben von wesentlichen Informationen in Zeichenmodellen, spricht man von Schemata. Hauptinstrumente sind Konstruktionsalgorithmen und Charakterisierungen von Graphenklassen, die für die Konstruktion geeignet sind. In dieser Arbeit werden Schemata für Graphen und Hypergraphen formalisiert und mit den genannten Instrumenten untersucht. In der Dissertation wird zunächst das „partial edge drawing“ (kurz: PED) Modell für Graphen (bezüglich gradliniger Zeichnung) untersucht. Dabei wird um Kreuzungen im Zentrum der Kante visuell zu eliminieren jede Kante durch ein kreuzungsfreies Teilstück (= Stummel) am Start- und am Zielknoten ersetzt. Als Standard hat sich eine PED-Variante etabliert, in der das Längenverhältnis zwischen Stummel und Kante genau 1⁄4 ist (kurz: 1⁄4-SHPED). Für 1⁄4-SHPEDs werden Konstruktionsalgorithmen, Klassifizierung, Implementierung und Evaluation präsentiert. Außerdem werden PED-Varianten mit festen Knotenpositionen und auf Basis orthogonaler Zeichnungen erforscht. Danach wird das BUS Modell für Hypergraphen untersucht, in welchem Hyperkanten durch fette horizontale oder vertikale – als BUS bezeichnete – Segmente repräsentiert werden. Dazu wird eine vollständige Charakterisierung von planaren Inzidenzgraphen von Hypergraphen angegeben, die eine planare Zeichnung im BUS Modell besitzen, und diverse planare BUS-Varianten mit festen Knotenpositionen werden diskutiert. Zum Schluss wird erstmals eine Punktmenge von subquadratischer Größe angegeben, die eine planare Einbettung (Knoten werden auf Punkte abgebildet) von 2-außenplanaren Graphen ermöglicht