9 research outputs found
Local Structure in the Web
posterInternational audienceThe web graph has been widely adopted as the core describing the web structure [4]. However, little attention has been paid to the relationship betweenthe web graph and the location of the pages. It has already been noticed that links are often local (i.e. from a page to another page of the same server) and this can be used for efficient encoding of the web graph [9,7]. Locality in the web can be further modelled by the clustered graph induced by the prefix tree of URLs. The web tree's internal nodes are the commonprefixes of URLs and its leaves are the URLs themselves. A prefix ordering of URLs according to this tree allows to observe local structure in the web directly on the adjacency matrix M of the web graph. M splits in two terms : M = D + S, where D is diagonal by blocks and S is a very sparse matrix. The blocks of D that can be observed along the diagonal are sets of pages strongly related together
Local Structure in the Web
posterInternational audienceThe web graph has been widely adopted as the core describing the web structure [4]. However, little attention has been paid to the relationship betweenthe web graph and the location of the pages. It has already been noticed that links are often local (i.e. from a page to another page of the same server) and this can be used for efficient encoding of the web graph [9,7]. Locality in the web can be further modelled by the clustered graph induced by the prefix tree of URLs. The web tree's internal nodes are the commonprefixes of URLs and its leaves are the URLs themselves. A prefix ordering of URLs according to this tree allows to observe local structure in the web directly on the adjacency matrix M of the web graph. M splits in two terms : M = D + S, where D is diagonal by blocks and S is a very sparse matrix. The blocks of D that can be observed along the diagonal are sets of pages strongly related together
Relaxing the Constraints of Clustered Planarity
In a drawing of a clustered graph vertices and edges are drawn as points and
curves, respectively, while clusters are represented by simple closed regions.
A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region,
or region-region crossings. Determining the complexity of testing whether a
clustered graph admits a c-planar drawing is a long-standing open problem in
the Graph Drawing research area. An obvious necessary condition for c-planarity
is the planarity of the graph underlying the clustered graph. However, such a
condition is not sufficient and the consequences on the problem due to the
requirement of not having edge-region and region-region crossings are not yet
fully understood.
In order to shed light on the c-planarity problem, we consider a relaxed
version of it, where some kinds of crossings (either edge-edge, edge-region, or
region-region) are allowed even if the underlying graph is planar. We
investigate the relationships among the minimum number of edge-edge,
edge-region, and region-region crossings for drawings of the same clustered
graph. Also, we consider drawings in which only crossings of one kind are
admitted. In this setting, we prove that drawings with only edge-edge or with
only edge-region crossings always exist, while drawings with only region-region
crossings may not. Further, we provide upper and lower bounds for the number of
such crossings. Finally, we give a polynomial-time algorithm to test whether a
drawing with only region-region crossings exist for biconnected graphs, hence
identifying a first non-trivial necessary condition for c-planarity that can be
tested in polynomial time for a noticeable class of graphs
Kreuzungen in Cluster-Level-Graphen
Clustered graphs are an enhanced graph model with a recursive clustering of the vertices according to a given nesting relation. This prime technique for expressing coherence of certain parts of the graph is used in many applications, such as biochemical pathways and UML class diagrams. For directed clustered graphs usually level drawings are used, leading to clustered level graphs. In this thesis we analyze the interrelation of clusters and levels and their influence on edge crossings and cluster/edge crossings.Cluster-Graphen sind ein erweitertes Graph-Modell mit einem rekursiven Clustering der Knoten entsprechend einer gegebenen Inklusionsrelation. Diese bedeutende Technik um Zusammengehörigkeit bestimmter Teile des Graphen auszudrĂŒcken wird in vielen Anwendungen benutzt, etwa biochemischen Reaktionsnetzen oder UML Klassendiagrammen. FĂŒr gerichtete Cluster-Graphen werden ĂŒblicherweise Level-Zeichnungen verwendet, was zu Cluster-Level-Graphen fĂŒhrt. Diese Arbeit analysiert den Zusammenhang zwischen Clustern und Level und deren Auswirkungen auf Kantenkreuzungen und Cluster/Kanten-Kreuzungen
New Approaches to Classic Graph-Embedding Problems - Orthogonal Drawings & Constrained Planarity
Drawings of graphs are often used to represent a given data set in a human-readable way. In this thesis, we consider different classic algorithmic problems that arise when automatically generating graph drawings. More specifically, we solve some open problems in the context of orthogonal drawings and advance the current state of research on the problems clustered planarity and simultaneous planarity
Interactive graph drawing with constraints
This thesis investigates the requirements for graph drawing stemming
from practical applications, and presents both theoretical as
well as practical results and approaches to handle them.
Many approaches to compute graph layouts in various drawing styles
exist, but the results are often not sufficient
for use in practice. Drawing conventions, graphical notation standards,
and user-defined requirements restrict the set of admissible
drawings. These restrictions can be formalized as constraints for the
layout computation. We investigate the requirements and give an overview
and categorization of the corresponding constraints.
Of main importance for the readability of a graph drawing is
the number of edge crossings. In case the graph is planar
it should be drawn without crossings, otherwise we should
aim to use the minimum number of crossings possible.
However, several types of constraints may impose
restrictions on the way the graph can be embedded in the plane.
These restrictions may have a strong impact on crossing minimization.
For two types of such constraints we present specific solutions
how to consider them in layout computation:
We introduce the class of so-called embedding constraints, which
restrict the order of the edges around a vertex.
For embedding constraints we describe approaches for planarity testing,
embedding, and edge insertion with the minimum number of crossings. These problems
can be solved in linear time with our approaches.
The second constraint type that we tackle are clusters. Clusters
describe a hierarchical grouping of the graph's vertices that
has to be reflected in the drawing. The complexity of the
corresponding clustered planarity testing problem for
clustered graphs is unknown so far.
We describe a technique to compute a maximum clustered planar
subgraph of a clustered graph. Our solution
is based on an Integer Linear Program (ILP) formulation and includes
also the first practical clustered planarity test for general clustered
graphs. The resulting subgraph can be used within the first step of
the planarization approach for clustered graphs.
In addition, we describe how to improve the performance
for pure clustered planarity testing by implying a branch-and-price
approach.
Large and complex graphs nowadays arise in many application domains.
These graphs require interaction
and navigation techniques to allow exploration of the underlying data.
The corresponding concepts are presented and solutions for three
practical applications are proposed: First, we describe Scaffold Hunter,
a tool for the exploration of chemical space. We show how to use
a hierarchical classification of molecules for the visual navigation in chemical space.
The resulting visualization is embedded into an interactive environment
that allows visual analysis of chemical compound databases.
Finally, two interactive
visualization approaches for two types of biological networks, protein-domain
networks and residue interaction networks, are presented.In zahlreichen Anwendungsgebieten werden Informationen als Graphen modelliert und
mithilfe dieser Graphen visualisiert. Eine ĂŒbersichtliche Darstellung hilft
bei der Analyse und unterstĂŒtzt das VerstĂ€ndnis
bei der PrÀsentation von Informationen mittels graph-basierter Diagramme.
Neben allgemeinen Ă€sthetischen Kriterien bestehen fĂŒr eine solche Darstellung
Anforderungen, die sich aus der Charakteristik der Daten, etablierten Darstellungskonventionen
und der konkreten Fragestellung ergeben. ZusÀtzlich ist hÀufig eine
individuelle Anpassung der Darstellung durch den Anwender gewĂŒnscht. Diese Anforderungen können mithilfe von Nebenbedingungen
fĂŒr die Berechnung eines Layouts formuliert werden.
Trotz einer Vielzahl unterschiedlicher Anforderungen aus zahlreichen Anwendungsgebieten können die meisten Anforderungen ĂŒber einige generische Nebenbedingungen formuliert werden.
In dieser Arbeit untersuchen wir die Anforderungen
aus der Praxis und beschreiben eine Zuordnung zu Nebenbedingungen fĂŒr
die Layoutberechnung. Wir geben eine Ăbersicht ĂŒber den aktuellen
Stand der Behandlung von Nebenbedingungen beim Zeichnen von Graphen
und kategorisieren diese nach grundlegenden Eigenschaften.
Von besonderer Wichtigkeit fĂŒr die QualitĂ€t einer Darstellung ist die
Anzahl der Kreuzungen. Planare Graphen sollten kreuzungsfrei gezeichnet
werden, bei nicht-planaren Graphen sollte die minimale Anzahl Kreuzungen
erreicht werden. Einige Nebenbedingungen beschrÀnken jedoch die
Möglichkeit, den Graph in die Ebene einzubetten. Dies kann starke
Auswirkungen auf das Ergebnis der Kreuzungsminimierung haben.
Zwei wichtige Typen solcher Nebenbedingungen werden in dieser Arbeit nÀher
untersucht. Mit den Embedding Constraints fĂŒhren wir eine Klasse
von Nebenbedingungen ein, welche die mögliche Reihenfolge der Kanten um einen Knoten
beschrĂ€nken. FĂŒr diese Klasse prĂ€sentieren wir Linearzeitalgorithmen
fĂŒr das Testen der PlanaritĂ€t und das optimale EinfĂŒgen von Kanten
unter Beachtung der EinbettungsbeschrÀnkungen.
Der zweite Typ von Nebenbedingungen sind Cluster, die eine hierarchische
Gruppierung von Knoten vorgeben. FĂŒr das Testen der Cluster-PlanaritĂ€t unter
solchen Nebenbedingungen ist die KomplexitÀt bisher unbekannt.
Wir beschreiben ein Verfahren, um einen maximalen Cluster-planaren
Untergraphen zu berechnen.
Wir nutzen dabei eine Formulierung als ganzzahliges lineares Programm
sowie einen Branch-and-Cut Ansatz zur Lösung. Das Verfahren erlaubt
auch die Bestimmung der Cluster-PlanaritÀt und
stellt damit den ersten praktischen Ansatz zum Testen
allgemeiner Clustergraphen dar. ZusÀtzlich
beschreiben wir eine Verbesserung fĂŒr den Fall, dass lediglich Cluster-PlanaritĂ€t
getestet werden muss, der maximale Cluster-planare Untergraph aber nicht
von Interesse ist. FĂŒr dieses Szenario geben wir eine vereinfachte Formulierung
und prÀsentieren ein Lösungsverfahren, das auf einem Branch-and-Price Ansatz beruht.
In der Praxis mĂŒssen hĂ€ufig sehr groĂe oder komplexe Graphen untersucht
werden. Dazu werden entsprechende Interaktions- und Navigationsmethoden
benötigt. Wir beschreiben die entsprechenden Konzepte und stellen Lösungen
fĂŒr drei Anwendungsbereiche vor:
ZunÀchst beschreiben wir Scaffold Hunter, eine Software zur Navigation
im chemischen Strukturraum. Scaffold Hunter benutzt eine hierarchische
Klassifikation von MolekĂŒlen als Grundlage fĂŒr die visuelle Navigation.
Die Visualisierung ist eingebettet in eine interaktive OberflÀche die
eine visuelle Analyse von chemischen Strukturdatenbanken erlaubt.
FĂŒr zwei Typen von biologischen Netzwerken, Protein-DomĂ€nen Netzwerke
und Residue-Interaktionsnetzwerke, stellen wir AnsĂ€tze fĂŒr die interaktive
Visualisierung dar. Die entsprechenden Layoutverfahren unterliegen einer
Reihe von Nebenbedingungen fĂŒr eine sinnvolle Darstellung