291 research outputs found
An Algorithmic Framework for Labeling Road Maps
Given an unlabeled road map, we consider, from an algorithmic perspective,
the cartographic problem to place non-overlapping road labels embedded in their
roads. We first decompose the road network into logically coherent road
sections, e.g., parts of roads between two junctions. Based on this
decomposition, we present and implement a new and versatile framework for
placing labels in road maps such that the number of labeled road sections is
maximized. In an experimental evaluation with road maps of 11 major cities we
show that our proposed labeling algorithm is both fast in practice and that it
reaches near-optimal solution quality, where optimal solutions are obtained by
mixed-integer linear programming. In comparison to the standard OpenStreetMap
renderer Mapnik, our algorithm labels 31% more road sections in average.Comment: extended version of a paper to appear at GIScience 201
Lombardi Drawings of Graphs
We introduce the notion of Lombardi graph drawings, named after the American
abstract artist Mark Lombardi. In these drawings, edges are represented as
circular arcs rather than as line segments or polylines, and the vertices have
perfect angular resolution: the edges are equally spaced around each vertex. We
describe algorithms for finding Lombardi drawings of regular graphs, graphs of
bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International
Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure
Achieving Good Angular Resolution in 3D Arc Diagrams
We study a three-dimensional analogue to the well-known graph visualization
approach known as arc diagrams. We provide several algorithms that achieve good
angular resolution for 3D arc diagrams, even for cases when the arcs must
project to a given 2D straight-line drawing of the input graph. Our methods
make use of various graph coloring algorithms, including an algorithm for a new
coloring problem, which we call localized edge coloring.Comment: 12 pages, 5 figures; to appear at the 21st International Symposium on
Graph Drawing (GD 2013
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
Curved Space-Filling Tiles Using Voronoi Decomposition with Line, and Curve Segments Closed Under Wallpaper Symmetries
In this paper, we present a new approach to obtain symmetric tiles with
curved edges. Our approach is based on using higher-order Voronoi sites that
are closed under wallpaper symmetries. The resulting Voronoi tessellations
provide us with symmetric tiles with curved edges. We have developed a web
application that provides real-time tile design. Our application can be found
at https://voronoi.viz.tamu.edu. One of our key findings in this paper is that
not all symmetry operations are useful for creating curved tiles. In
particular, all symmetries that use mirror operation produce straight lines
that are useless for creating new tiles. This result is interesting because it
suggests that we need to avoid mirror transformations to produce unusual
space-filling tiles in 2D and 3D using Voronoi tessellations.Comment: 1
An Algorithmic Framework for Labeling Network Maps
Drawing network maps automatically comprises two challenging steps, namely
laying out the map and placing non-overlapping labels. In this paper we tackle
the problem of labeling an already existing network map considering the
application of metro maps. We present a flexible and versatile labeling model.
Despite its simplicity, we prove that it is NP-complete to label a single line
of the network. For a restricted variant of that model, we then introduce an
efficient algorithm that optimally labels a single line with respect to a given
weighting function. Based on that algorithm, we present a general and
sophisticated workflow for multiple metro lines, which is experimentally
evaluated on real-world metro maps.Comment: Full version of COCOON 2015 pape
CelticGraph: Drawing Graphs as Celtic Knots and Links
Celtic knots are an ancient art form often attributed to Celtic cultures,
used to decorate monuments and manuscripts, and to symbolise eternity and
interconnectedness. This paper describes the framework CelticGraph to draw
graphs as Celtic knots and links. The drawing process raises interesting
combinatorial concepts in the theory of circuits in planar graphs. Further,
CelticGraph uses a novel algorithm to represent edges as B\'ezier curves,
aiming to show each link as a smooth curve with limited curvature.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
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