253 research outputs found
Drawing Order Diagrams Through Two-Dimension Extension
Order diagrams are an important tool to visualize the complex structure of
ordered sets. Favorable drawings of order diagrams, i.e., easily readable for
humans, are hard to come by, even for small ordered sets. Many attempts were
made to transfer classical graph drawing approaches to order diagrams. Although
these methods produce satisfying results for some ordered sets, they
unfortunately perform poorly in general. In this work we present the novel
algorithm DimDraw to draw order diagrams. This algorithm is based on a relation
between the dimension of an ordered set and the bipartiteness of a
corresponding graph.Comment: 16 pages, 12 Figure
Extended high dimensional indexing approach for reachability queries on very large graphs
Given a directed acyclic graph G = (V,A) and two vertices u, v ∈ V , the reachability problem is to answer if there is a path from u to v in the graph. In the context of very large graphs, with millions of vertices and a series of queries to be answered, it is not practical to search the graph for each query. On the other hand, the storage of the full transitive closure of the graph is also impractical due to its O(|V |2) size. Scalable approaches aim to create indices used to prune the search during its execution. Negative indices may be able to determine (in constant time) that a query has a negative answer while positive indices may determine (again in constant time) that a query has a positive answer. In this paper we propose novel scalable approach called LYNX that uses a large number of topological sorts of G as a negative cut index without degrading the query time. A similar strategy is applied regarding a positive cut index. In addition, LYNX proposes a user-defined index size that enables the user to control the ratio between negative and positive cuts depending on the expected query pattern. We show by computational experiments that LYNX consistently outperforms the state-of-the-art approach in terms of query-time using the same index-size for graphs with high reachability ratio. In intelligent computer systems that rely on frequent tests of connectivity in graphs, LYNX can reduce the time delay experience by end users through a reduced query time. This comes at the expense of an increased setup time whenever the underlying graph is updated. Keywords: directed acyclic graphs, topological sorts, reachability queries, graph indexingpublishedVersio
Superpatterns and Universal Point Sets
An old open problem in graph drawing asks for the size of a universal point
set, a set of points that can be used as vertices for straight-line drawings of
all n-vertex planar graphs. We connect this problem to the theory of
permutation patterns, where another open problem concerns the size of
superpatterns, permutations that contain all patterns of a given size. We
generalize superpatterns to classes of permutations determined by forbidden
patterns, and we construct superpatterns of size n^2/4 + Theta(n) for the
213-avoiding permutations, half the size of known superpatterns for
unconstrained permutations. We use our superpatterns to construct universal
point sets of size n^2/4 - Theta(n), smaller than the previous bound by a 9/16
factor. We prove that every proper subclass of the 213-avoiding permutations
has superpatterns of size O(n log^O(1) n), which we use to prove that the
planar graphs of bounded pathwidth have near-linear universal point sets.Comment: GD 2013 special issue of JGA
The Partial Visibility Representation Extension Problem
For a graph , a function is called a \emph{bar visibility
representation} of when for each vertex , is a
horizontal line segment (\emph{bar}) and iff there is an
unobstructed, vertical, -wide line of sight between and
. Graphs admitting such representations are well understood (via
simple characterizations) and recognizable in linear time. For a directed graph
, a bar visibility representation of , additionally, puts the bar
strictly below the bar for each directed edge of
. We study a generalization of the recognition problem where a function
defined on a subset of is given and the question is whether
there is a bar visibility representation of with for every . We show that for undirected graphs this problem
together with closely related problems are \NP-complete, but for certain cases
involving directed graphs it is solvable in polynomial time.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Algorithms for Incremental Planar Graph Drawing and Two-page Book Embeddings
Subject of this work are two problems related to ordering the vertices
of planar graphs. The first one is concerned with the properties of
vertex-orderings that serve as a basis for incremental drawing algorithms.
Such a drawing algorithm usually extends a drawing by adding the vertices
step-by-step as provided by the ordering. In the field of graph drawing
several orderings are in use for this purpose. Some of them, however,
lack certain properties that are desirable or required for classic
incremental drawing methods. We narrow down these properties, and
introduce the bitonic st-ordering, an ordering which combines the
features only available when using canonical orderings with the flexibility
of st-orderings. The additional property of being bitonic enables an
st-ordering to be used in algorithms that usually require a canonical
ordering.
With this in mind, we describe a linear-time algorithm that computes
such an ordering for every biconnected planar graph. Unlike canonical
orderings, st-orderings extend to directed graphs, in particular planar
st-graphs. Being able to compute bitonic st-orderings for planar st-graphs
is of particular interest for upward planar drawing algorithms, since
traditional incremental algorithms for undirected planar graphs might
be adapted to directed graphs. Based on this observation, we give a
full characterization of the class of planar st-graphs that admit such
an ordering. This includes a linear-time algorithm for recognition
and ordering. Furthermore, we show that by splitting specific edges of
an instance that is not part of this class, one is able to transform
it into one for which then such an ordering exists. To do so, we describe
a linear-time algorithm for finding the smallest set of edges to split.
We show that for a planar st-graph G=(V,E), |V|−3 edge splits
are sufficient and every edge is split at most once. This immediately
translates to the number of bends required for upward planar poly-line
drawings. More specifically, we show that every planar st-graph admits
an upward planar poly-line drawing in quadratic area with at most |V|−3
bends in total and at most one bend per edge. Moreover, the drawing
can be obtained in linear time.
The second part is concerned with embedding planar graphs with maximum
degree three and four into books. Besides providing a simplified
incremental linear-time algorithm for embedding triconnected 3-planar
graphs into a book of two pages, we describe a linear-time algorithm
to compute a subhamiltonian cycle in a triconnected 4-planar graph
Upward planarization and layout
Die Visualisierung von gerichteten azyklischen Graphen (DAGs) gehört zu
den wichtigsten Aufgaben im automatischen Zeichnen von Graphen. Hierbei
suchen wir für einen gegebenen DAG G eine Zeichnung von G (Aufwärtszeichnung
von G genannt), sodass alle Kanten als Kurven streng monoton
in vertikaler Richtung steigend gezeichnet werden. Um die Lesbarkeit der
Zeichnung zu erhöhen, sollte neben der Aufwärtseigenschaft auch die Anzahl
der Kantenkreuzungen in der Zeichnung möglichst gering sein.
In dieser Dissertation entwerfen wir einen neuen Ansatz zur Visualisierung
von gerichteten Graphen, der auf der Idee der Aufwärtsplanarisierung basiert.
Wir stellen zuerst ein innovatives Aufwärtsplanarisierungverfahren vor, das
neue Techniken für die Berechnung aufwärtsplanare Untergraphen und die
anschließende Kanteneinfügephase einsetzt. Vor allem werden in dem neuen
Verfahren keine Schichtungstechniken zur Kreuzungsminimierung benutzt,
wie wir sie aus dem Zeichenverfahren von Sugiyama et al. [STT81] oder aus
dem Aufwärtsplanarisierungsverfahren von Eiglsperger et al. [EKE03] kennen.
Die Festlegung einer Schichtung kann nämlich zu sehr schlechten Ergebnissen
führen. Folglich besitzt das neue Verfahren nicht die Nachteile der bisherigen
Kreuzungsminimierungsverfahren.
Experimentellen Analysen zeigen, dass das neue Aufwärtsplanarisierungsverfahren
deutlich bessere Ergebnisse liefert als das klassische, auf Schichtungen
basierende Kreuzungsminimierungsverfahren, und dies unabhängig
von den benutzten Lösungsansätzen (heuristisch oder optimal) für die klevel
Kreuzungsminimierungsphase. Auch im Vergleich mit den bekannten
Aufwärtsplanarisierungsverfahren (Di Battista et al. [BPTT89] und Eiglsperger
et al. [EKE03]) zeigt sich, dass der neue Ansatz weitaus bessere Ergebnisse
liefert. Wir stellen auch zwei Erweiterungen des neuen Ansatzes vor:
eine Erweiterung zur Aufwärtsplanarisierung von gerichteten Hypergraphen
und eine zur Unterstützung von Port Constraints.
Das Ergebnis der Aufwärtsplanarisierung ist eine aufwärtsplanare Repräsentation (UPR) — ein eingebetteter DAG, in dem Kreuzungen durch
künstliche Dummy-Knoten modelliert werden. Wir stellen ein Layoutverfahren
zur Realisierung solcher UPRs vor, d.h., ein Verfahren, das aus einem
UPR eine Aufwärtszeichnung konstruiert, sodass die Kantenkreuzungen in
der Zeichnung zu den Dummy-Knoten des gegebenen UPR korrespondieren.
Die wenigen existierenden Zeichenverfahren zur Realisierung von UPRs sind
sehr einfach und wurden ursprünglich entwickelt, um planare st-Graphen zu
zeichnen. Unser neues Verfahren stellt somit das erste Layoutverfahren dar,
das speziell im Hinblick auf die Realisierung von UPRs entworfen wurde. Es
bietet zwei wichtige Vorteile gegenüber dem etablierten Standardzeichenalgorithmus
von Sugiyama et al.: Die Zeichnungen besitzen wesentlich weniger
Kreuzungen, was zur deutlichen Verbesserung der Lesbarkeit führt. Ferner
sind sie strukturierter und machen einen aufgeräumteren Eindruck
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