1,118 research outputs found
More compact orthogonal drawings by allowing additional bends
Compacting orthogonal drawings is a challenging task. Usually, algorithms try to compute drawings with small area or total edge length while preserving the underlying orthogonal shape. We suggest a moderate relaxation of the orthogonal compaction problem, namely the one-dimensional monotone flexible edge compaction problem with fixed vertex star geometry. We further show that this problem can be solved in polynomial time using a network flow model. An experimental evaluation shows that by allowing additional bends could reduce the total edge length and the drawing area
Stress-Minimizing Orthogonal Layout of Data Flow Diagrams with Ports
We present a fundamentally different approach to orthogonal layout of data
flow diagrams with ports. This is based on extending constrained stress
majorization to cater for ports and flow layout. Because we are minimizing
stress we are able to better display global structure, as measured by several
criteria such as stress, edge-length variance, and aspect ratio. Compared to
the layered approach, our layouts tend to exhibit symmetries, and eliminate
inter-layer whitespace, making the diagrams more compact
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
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
On Compact RAC Drawings
We present new bounds for the required area of Right Angle Crossing (RAC) drawings for complete graphs, i.e. drawings where any two crossing edges are perpendicular to each other. First, we improve upon results by Didimo et al. [Walter Didimo et al., 2011] and Di Giacomo et al. [Emilio Di Giacomo et al., 2011] by showing how to compute a RAC drawing with three bends per edge in cubic area. We also show that quadratic area can be achieved when allowing eight bends per edge in general or with three bends per edge for p-partite graphs. As a counterpart, we prove that in general quadratic area is not sufficient for RAC drawings with three bends per edge
Ortho-Radial Drawing in Near-Linear Time
An orthogonal drawing is an embedding of a plane graph into a grid. In a seminal work of Tamassia (SIAM Journal on Computing 1987), a simple combinatorial characterization of angle assignments that can be realized as bend-free orthogonal drawings was established, thereby allowing an orthogonal drawing to be described combinatorially by listing the angles of all corners. The characterization reduces the need to consider certain geometric aspects, such as edge lengths and vertex coordinates, and simplifies the task of graph drawing algorithm design.
Barth, Niedermann, Rutter, and Wolf (SoCG 2017) established an analogous combinatorial characterization for ortho-radial drawings, which are a generalization of orthogonal drawings to cylindrical grids. The proof of the characterization is existential and does not result in an efficient algorithm. Niedermann, Rutter, and Wolf (SoCG 2019) later addressed this issue by developing quadratic-time algorithms for both testing the realizability of a given angle assignment as an ortho-radial drawing without bends and constructing such a drawing.
In this paper, we improve the time complexity of these tasks to near-linear time. We establish a new characterization for ortho-radial drawings based on the concept of a good sequence. Using the new characterization, we design a simple greedy algorithm for constructing ortho-radial drawings
Axis-Parallel Right Angle Crossing Graphs
A RAC graph is one admitting a RAC drawing, that is, a polyline drawing in which each crossing occurs at a right angle. Originally motivated by psychological studies on readability of graph layouts, RAC graphs form one of the most prominent graph classes in beyond planarity.
In this work, we study a subclass of RAC graphs, called axis-parallel RAC (or apRAC, for short), that restricts the crossings to pairs of axis-parallel edge-segments. apRAC drawings combine the readability of planar drawings with the clarity of (non-planar) orthogonal drawings. We consider these graphs both with and without bends. Our contribution is as follows: (i) We study inclusion relationships between apRAC and traditional RAC graphs. (ii) We establish bounds on the edge density of apRAC graphs. (iii) We show that every graph with maximum degree 8 is 2-bend apRAC and give a linear time drawing algorithm. Some of our results on apRAC graphs also improve the state of the art for general RAC graphs. We conclude our work with a list of open questions and a discussion of a natural generalization of the apRAC model
Small Superpatterns for Dominance Drawing
We exploit the connection between dominance drawings of directed acyclic
graphs and permutations, in both directions, to provide improved bounds on the
size of universal point sets for certain types of dominance drawing and on
superpatterns for certain natural classes of permutations. In particular we
show that there exist universal point sets for dominance drawings of the Hasse
diagrams of width-two partial orders of size O(n^{3/2}), universal point sets
for dominance drawings of st-outerplanar graphs of size O(n\log n), and
universal point sets for dominance drawings of directed trees of size O(n^2).
We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}),
riffle permutations (321-, 2143-, and 2413-avoiding permutations) have
superpatterns of size O(n), and the concatenations of sequences of riffles and
their inverses have superpatterns of size O(n\log n). Our analysis includes a
calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of
the 321-superpattern siz
Axis-Parallel Right Angle Crossing Graphs
A RAC graph is one admitting a RAC drawing, that is, a polyline drawing in
which each crossing occurs at a right angle. Originally motivated by
psychological studies on readability of graph layouts, RAC graphs form one of
the most prominent graph classes in beyond planarity.
In this work, we study a subclass of RAC graphs, called axis-parallel RAC (or
apRAC, for short), that restricts the crossings to pairs of axis-parallel
edge-segments. apRAC drawings combine the readability of planar drawings with
the clarity of (non-planar) orthogonal drawings. We consider these graphs both
with and without bends. Our contribution is as follows: (i) We study inclusion
relationships between apRAC and traditional RAC graphs. (ii) We establish
bounds on the edge density of apRAC graphs. (iii) We show that every graph with
maximum degree 8 is 2-bend apRAC and give a linear time drawing algorithm. Some
of our results on apRAC graphs also improve the state of the art for general
RAC graphs. We conclude our work with a list of open questions and a discussion
of a natural generalization of the apRAC model
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