Convex-Arc Drawings of Pseudolines ⋆

Introduction. A pseudoline is formed from a line by stretching the plane without tearing: it is the image of a line under a homeomorphism of the plane [13]. In arrangements of pseudolines, pairs of pseudolines intersect at most once and cross at their intersections. Pseudoline arrangements can be used to model sorting networks [1], tilings of convex polygons by rhombi [4], and graphs that have distance-preserving embedding

Convex-Arc Drawings of Pseudolines

A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar---each crossing is incident to an unbounded face---and simple---each crossing point is the crossing of only two curves. We show that these arrangements can be represented by chords of a circle, by convex polygonal chains with only two bends, or by hyperbolic lines. Simple but non-outerplanar arrangements (non-weak) can be represented by convex polygonal chains or convex smooth curves of linear complexity.Comment: 11 pages, 8 figures. A preliminary announcement of these results was made as a poster at the 21st International Symposium on Graph Drawing, Bordeaux, France, September 2013, and published in Lecture Notes in Computer Science 8242, Springer, 2013, pp. 522--52

Short Plane Supports for Spatial Hypergraphs

A graph $G=(V,E)$ is a support of a hypergraph $H=(V,S)$ if every hyperedge induces a connected subgraph in $G$. 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 $V$. 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

Pictures of the Past : Visualization and visual analysis in archaeological context

Data visualization, the main topic of this dissertation, is the science concerned with the design and creation of visual representations intended to convey information and facilitate understanding. In a scientific context, the data to be visualized are typically research results, which can originate from any discipline. The main challenges in data visualization are to find visual representations which (1) are as accurate as possible, (2) can be understood quickly and well by viewers, and (3) can be produced automatically and efficiently. The latter is especially important when large or dynamic data sets are involved. This dissertation is focused on addressing these challenges for a particular application domain: the visualization of archaeological data. We identify general properties that often characterize archaeological data and discuss a variety of visualization methods suitable for data with these properties. For existing methods from the area of mathematics and computer science, we explain how they can be applied – either directly or with some adaptations – in the context of archaeology. We also introduce new approaches, tailored to various archaeological applications. Case studies and real archaeological data sets are used to demonstrate the use of these approaches in practical applications. However, because of our focus on general data properties rather than specific data sets, the methods presented in this dissertation are in fact widely applicable beyond the field of archaeology as well.publishe

A Labeling Problem for Symbol Maps of Archaeological Sites

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Minimum-Displacement Overlap Removal for Geo-referenced Data Visualization

Given a set of rectangles embedded in the plane, we consider the problem of adjusting the layout to remove all overlap while preserving the orthogonal order of the rectangles. The objective is to minimize the displacement of the rectangles. We call this problem MINIMUM-DISPLACEMENT OVERLAP REMOVAL (MDOR). Our interest in this problem is motivated by the application of displaying metadata of archaeological sites. Because most existing overlap removal algorithms are not designed to minimize displacement while preserving orthogonal order, we present and compare several approaches which are tailored to our particular usecase. We introduce a new overlap removal heuristic which we call REARRANGE. Although conceptually simple, it is very effective in removing the overlap while keeping the displacement small. Furthermore, we propose an additional procedure to repair the orthogonal order after every iteration, with which we extend both our new heuristic and PRISM, a widely used overlap removal algorithm. We compare the performance of both approaches with and without this order repair method. The experimental results indicate that REARRANGE is very effective for heterogeneous input data where the overlap is concentrated in few dense regions.publishe

Recognizing a DOG is Hard but not when it is Thin and Unit

We define the notion of disk-obedience for a set of disks in the plane and give results for disk-obedient graphs (DOGs), which are disk intersection graphs (DIGs) that admit a planar embedding with vertices inside the corresponding disks. We show that in general it is hard to recognize a DOG, but when the DIG is thin and unit (i.e., when the disks are unit disks), it can be done in linear time

On graphical representations of similarity in geo-temporal frequency data

Its focus on dependencies and patterns in relational data makes network science a promising addition to the analytic toolbox in archaeology. Despite its tradition in a number of other fields, however, the methodology of network science is only in development and its scope and proper usage are subject to debate. We argue that the historical linkage with graph theory and limitations in commonly available software form an obstacle to leveraging the full potential of network methods. This is illustrated via replication of a study of Maya obsidian (Golitko et al. Antiquity, 2012), in which it seemed necessary to discard detailed information in order to represent data in networks suitable for further processing. We propose means to avoid such information loss by using methods capable of handling valued rather than binarized data. The resulting representations corroborate previous conclusions but are more reliable and thus justify a more detailed interpretation of shifting supply routes as an underlying process contributing to the collapse of Maya urban centers. Some general conclusions for the use of network science in archaeology are offered.publishe

Convex-arc drawings of pseudolines

A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar---each crossing is incident to an unbounded face---and simple---each crossing point is the crossing of only two curves. We show that these arrangements can be represented by chords of a circle, by convex polygonal chains with only two bends, or by hyperbolic lines. Simple but non-outerplanar arrangements (non-weak) can be represented by convex polygonal chains or convex smooth curves of linear complexity