47 research outputs found
The role of twins in computing planar supports of hypergraphs
A support or realization of a hypergraph is a graph on the same
vertex as such that for each hyperedge of it holds that its vertices
induce a connected subgraph of . The NP-hard problem of finding a planar}
support has applications in hypergraph drawing and network design. Previous
algorithms for the problem assume that twins}---pairs of vertices that are in
precisely the same hyperedges---can safely be removed from the input
hypergraph. We prove that this assumption is generally wrong, yet that the
number of twins necessary for a hypergraph to have a planar support only
depends on its number of hyperedges. We give an explicit upper bound on the
number of twins necessary for a hypergraph with hyperedges to have an
-outerplanar support, which depends only on and . Since all
additional twins can be safely removed, we obtain a linear-time algorithm for
computing -outerplanar supports for hypergraphs with hyperedges if
and are constant; in other words, the problem is fixed-parameter
linear-time solvable with respect to the parameters and
On Embeddability of Buses in Point Sets
Set membership of points in the plane can be visualized by connecting
corresponding points via graphical features, like paths, trees, polygons,
ellipses. In this paper we study the \emph{bus embeddability problem} (BEP):
given a set of colored points we ask whether there exists a planar realization
with one horizontal straight-line segment per color, called bus, such that all
points with the same color are connected with vertical line segments to their
bus. We present an ILP and an FPT algorithm for the general problem. For
restricted versions of this problem, such as when the relative order of buses
is predefined, or when a bus must be placed above all its points, we provide
efficient algorithms. We show that another restricted version of the problem
can be solved using 2-stack pushall sorting. On the negative side we prove the
NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201
Short Plane Supports for Spatial Hypergraphs
A graph is a support of a hypergraph if every hyperedge
induces a connected subgraph in . Supports are used for certain types of
hypergraph visualizations. In this paper we consider visualizing spatial
hypergraphs, where each vertex has a fixed location in the plane. This is the
case, e.g., when modeling set systems of geospatial locations as hypergraphs.
By applying established aesthetic quality criteria we are interested in finding
supports that yield plane straight-line drawings with minimum total edge length
on the input point set . We first show, from a theoretical point of view,
that the problem is NP-hard already under rather mild conditions as well as a
negative approximability results. Therefore, the main focus of the paper lies
on practical heuristic algorithms as well as an exact, ILP-based approach for
computing short plane supports. We report results from computational
experiments that investigate the effect of requiring planarity and acyclicity
on the resulting support length. Further, we evaluate the performance and
trade-offs between solution quality and speed of several heuristics relative to
each other and compared to optimal solutions.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
MetroSets: Visualizing Sets as Metro Maps
We propose MetroSets, a new, flexible online tool for visualizing set systems
using the metro map metaphor. We model a given set system as a hypergraph
, consisting of a set of vertices and a set
, which contains subsets of called hyperedges. Our system then
computes a metro map representation of , where each hyperedge
in corresponds to a metro line and each vertex corresponds to a
metro station. Vertices that appear in two or more hyperedges are drawn as
interchanges in the metro map, connecting the different sets. MetroSets is
based on a modular 4-step pipeline which constructs and optimizes a path-based
hypergraph support, which is then drawn and schematized using metro map layout
algorithms. We propose and implement multiple algorithms for each step of the
MetroSet pipeline and provide a functional prototype with \new{easy-to-use
preset configurations.} % many real-world datasets. Furthermore, \new{using
several real-world datasets}, we perform an extensive quantitative evaluation
of the impact of different pipeline stages on desirable properties of the
generated maps, such as octolinearity, monotonicity, and edge uniformity.Comment: 19 pages; accepted for IEEE INFOVIS 2020; for associated live system,
see http://metrosets.ac.tuwien.ac.a
Constraint Generation Algorithm for the Minimum Connectivity Inference Problem
Given a hypergraph , the Minimum Connectivity Inference problem asks for a
graph on the same vertex set as with the minimum number of edges such that
the subgraph induced by every hyperedge of is connected. This problem has
received a lot of attention these recent years, both from a theoretical and
practical perspective, leading to several implemented approximation, greedy and
heuristic algorithms. Concerning exact algorithms, only Mixed Integer Linear
Programming (MILP) formulations have been experimented, all representing
connectivity constraints by the means of graph flows. In this work, we
investigate the efficiency of a constraint generation algorithm, where we
iteratively add cut constraints to a simple ILP until a feasible (and optimal)
solution is found. It turns out that our method is faster than the previous
best flow-based MILP algorithm on random generated instances, which suggests
that a constraint generation approach might be also useful for other
optimization problems dealing with connectivity constraints. At last, we
present the results of an enumeration algorithm for the problem.Comment: 16 pages, 4 tables, 1 figur
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
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Formalizing graphical notations
The thesis describes research into graphical notations for software engineering, with a principal interest in ways of formalizing them. The research seeks to provide a theoretical basis that will help in designing both notations and the software tools that process them.
The work starts from a survey of literature on notation, followed by a review of techniques for formal description and for computational handling of notations. The survey concentrates on collecting views of the benefits and the problems attending notation use in software development; the review covers picture description languages, grammars and tools such as generic editors and visual programming environments. The main problem of notation is found to be a lack of any coherent, rigorous description methods. The current approaches to this problem are analysed as lacking in consensus on syntax specification and also lacking a clear focus on a defined concept of notated expression.
To address these deficiencies, the thesis embarks upon an exploration of serniotic, linguistic and logical theory; this culminates in a proposed formalization of serniosis in notations, using categorial model theory as a mathematical foundation. An argument about the structure of sign systems leads to an analysis of notation into a layered system of tractable theories, spanning the gap between expressive pictorial medium and subject domain. This notion of 'tectonic' theory aims to treat both diagrams and formulae together.
The research gives details of how syntactic structure can be sketched in a mathematical sense, with examples applying to software development diagrams, offering a new solution to the problem of notation specification. Based on these methods, the thesis discusses directions for resolving the harder problems of supporting notation design, processing and computer-aided generic editing. A number of future research areas are thereby opened up. For practical trial of the ideas, the work proceeds to the development and partial implementation of a system to aid the design of notations and editors. Finally the thesis is evaluated as a contribution to theory in an area which has not attracted a standard approach
System Modeling and Traceability Applications of the Higraph Formalism
One of the most important tools for a systems engineer is their system model. From this model, engineering decisions can be made without costly integration, fabrication, or installations. Existing system modeling languages used to create the system model are detailed and comprehensive, but lack a true ability to unify the system model by showing all relationships among all components in the model. Higraphs, a type of mathematical graph, allow systems engineers to not only represent all required information in a system model, but to formally show all relationships in the model through hierarchies, edges, and orthogonalities. With a higraph system model, all relationships between system requirements, components, and behaviors are formalized allowing for a "smart" model that can be queried for custom sets of information that, when presented to the systems engineer, will aid in engineering decisions
Applying the Engineering Statechart Formalism to the evaluation of soft real-time in operating systems : a use case tailored modeling and analysis technique
Multimedia applications that have emerged in recent years impose unique requirements on an underlying general purpose operating system (GPOS). The suitability of a GPOS for multimedia processing is judged by its soft real-time capabilities. To date, the question of how these capabilities can be assessed has scarcely been addressed: this is a gap in GPOS research. By answering questions on the impacts of the Interrupt Handling Facility (IHF) on the overall soft real-time capabilities of a GPOS, this thesis contributes to the filling of this blank space. The Engineering Statechart Formalism (ESF), a use case tailored formal method of modeling real-world OS, is syntactically and semantically defined. Models of the IHF of selected real-world operating systems are then created by means of this technique. As no appropriate real-time concept fitting the goals of this thesis as yet exists, a suitable definition is constructed. By projecting this system-wide idea to the interrupt subsystem, specific indicators for this subsystem are erived. These indicators are then evaluated by applying formal techniques such as graph-based analysis and temporal logic model checking to the ESF models. Finally, the assertions derived from this evaluation are interpreted with respect to their impacts on real-time multimedia processing in different general purpose operating systems.Multimedia-Anwendungen haben in den letzten Jahren weite Verbreitung erfahren. Solche Anwendungen stellen besondere Anforderungen an das Betriebssystem (BS), auf dem sie ausgeführt werden. Insbesondere Echtzeitfähigkeiten des Betriebssystems sind von Bedeutung, wenn es um seine Eignung für Multimedia-Verarbeitung geht. Bis heute wurde die Frage, wie sich diese Fähigkeiten konkret innerhalb eines BS manifestieren, nur unzureichend untersucht. Die vorliegende Arbeit leistet einen Beitrag zur Füllung dieser Lücke in der BS-Forschung. Die Effekte des Subsystems zur Unterbrechungsbehandlung in BS auf die Echtzeitfähigkeit des Gesamtsystems werden detailliert auf Basis von Modellen dieses Subsystems in verschiedenen BS analysiert. Um eine formale Auswertung zu erlauben, wird eine auf den Anwendungsfall zugeschnittene formale Methode zur BS-Modellierung verwendet. Die spezifizierte Syntax und Semantik dieses Engineering Statechart Formalism (ESF) basieren auf dem klassischen Statechart-Formalismus. Da bislang kein geeigneter Echtzeit-Begriff existiert, wird eine konsistente Definition hergeleitet. Durch die Abbildung dieser sich auf das Gesamtsystem beziehenden Eigenschaft auf die Unterbrechungsbehandlung werden spezifische Indikatoren für dieses Subsystem hergeleitet. Die Ausprägungen dieser Indikatoren für die verschiedenen untersuchten Betriebssyteme werden anhand formaler Methoden wie graphbasierter Analyse und Temporal Logic Model Checking ausgewertet. Die Interpretation der Untersuchungsergebnisse liefert Aussagen über die Effekte der Implementierung der Unterbrechungsbehandlung auf die Echtzeitfähigkeit der untersuchten Betriebssysteme bei der Verarbeitung von multimedialen Daten
Communities in temporal networks: from theoretical underpinnings to real-life applications
Static aggregations of network activity can unravel attributes of the complex systems they represent. However, they fall short when the structure of the systems changes over time. In some cases, changes are sluggish, such as in power grids, where lines enjoy a lengthy temporal permanence. In others, a high frequency of change is observed, such as on a network of online messages, social contacts, pathogen transmission or ball passing in a soccer game. In these cases, reducing what is inherently a temporal network to a static one, leads necessarily to a loss of information, such as causal relationships, precedence or reachability rules. Temporal networks are thus the main subject of this thesis, centered on the study of network evolution from the point of view of its clusters as significant meso-structures. The thesis has two interrelated parts. In the first, theoretical challenges are addressed and original algorithms, methods and tools are developed that can further the study of network theory. In the second, these developments are applied to the analysis of team invasion sports. A measurement of game dynamics was created based on a temporal network representation of a match, with nodes clustered by spatial proximity. These measurements were found to correlate with match events of known dynamics. Moreover, they reveal unique, multi-level, aspects of the game, from the individual players contributions, to the clusters of interacting players, to their teams and their matches, which is useful for game analysis, training and strategy development.As agregações estáticas das ligações de uma rede podem revelar atributos dos sistemas complexos que representam. Todavia, são insuficientes quando a estrutura dos sistemas se altera com o tempo. Em alguns casos, as transformações são lentas, tais como em redes de transmissão de eletricidade em que as linhas se mantêm inalteráveis por largos períodos de tempo. Noutras, regista-se uma taxa elevada de mudança, como por exemplo numa rede de mensagens em linha, contatos sociais, transmissão de patógenos ou passes num jogo de futebol. Nestes casos, reduzir o que é inerentemente uma rede temporal a uma rede estática, leva a uma perda de informação, tais como relações causais, regras de precedência ou de acessibilidade. Redes temporais são assim o tema desta tese, centrada nos seus agrupamentos, como meso-estruturas significantes. A tese está dividida em duas partes. Na primeira, são considerados problemas teóricos, e são desenvolvidos algoritmos, métodos e ferramentas que avançam o estudo da teoria de redes. Na segunda, estes desenvolvimentos são aplicados à análise de jogos desportivos coletivos de invasão. Foi criada uma medida de dinâmica do jogo, baseada na representação da partida através de uma rede temporal de nós agrupados por proximidade espacial. Os resultados obtidos correlacionam-se com eventos do jogo de dinâmica conhecida. Adicionalmente, esta medida revela aspetos únicos e multi-nível da dinâmica do jogo, desde a contribuição individual do jogador, até aos agrupamentos de jogadores, da equipa e das partidas, útil para a análise do jogo, de treino e de desenvolvimento estratégico