Publications

Please also see my DBLP and Google Scholar profiles.

Chapters in Books
Journals
Formally Reviewed Conference Proceedings
Weakly Refereed Conference Proceedings and Workshop Abstracts
Theses

2023

A Subquadratic Nε-Approximation for the Continuous Fréchet Distance.
Thijs van der Horst, Marc J. van Kreveld, Tim Ophelders, and Bettina Speckmann.
Proc. 34th Annual Symposium on Discrete Algorithms (SODA), pp. 1759—1776, 2023.

－ BibTeX

@article{a-subquadratic-n-sup-sup-approximation-for-the-continuous-fr-chet-distance-:2023,
    title = {A Subquadratic N^ε-Approximation for the Continuous Fréchet Distance.},
    author = {Thijs van der Horst and Marc J. van Kreveld and Tim Ophelders and Bettina Speckmann},
    year = {2023},
    bookTitle = {Proc. 34th Annual Symposium on Discrete Algorithms (SODA)},
}

Density Approximation for Kinetic Groups
Max J.M. van Mulken, Bettina Speckmann, and Kevin A.B. Verbeek.
pp. 16:1—16:7, 2023.

－ BibTeX

@article{density-approximation-for-kinetic-groups:2023,
    title = {Density Approximation for Kinetic Groups},
    author = {Max J.M. van Mulken and Bettina Speckmann and Kevin A.B. Verbeek},
    year = {2023},
    bookTitle = {},
}

2022

Agglomerative Clustering of Growing Squares
Thom Castermans, Bettina Speckmann, Frank Staals, and Kevin Verbeek.
Algorithmica, 84(1):216—233, 2022.

－ Abstract

We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of n disjoint growing squares. Our data structure uses O(nlog nlog log n) space, supports queries in worst case O(log 2n) time, and updates in O(log 5n) amortized time. This leads to an O(nα(n)log5n) time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best O(n2) time algorithms.
－ BibTeX
```
@article{agglomerative-clustering-of-growing-squares:2022,
 title = {Agglomerative Clustering of Growing Squares},
 author = {Thom Castermans and Bettina Speckmann and Frank Staals and Kevin Verbeek},
 year = {2022},
 bookTitle = {Algorithmica},
}
```
[ PDF ]
Alluvial Connectivity in Multi-Channel Networks in Rivers and Estuaries
Willem Sonke, Maarten G. Kleinhans, Bettina Speckmann, Wout M. van Dijk, and Matthew Hiatt.
Earth Surface Processes and Landforms, 47(2):477—490, 2022.

－ Abstract

Channels in rivers and estuaries are the main paths of fluvial and tidal currents that transport sediment through the system. While network representations of multi-channel systems and their connectivity are quite useful for characterisation of braiding patterns and dynamics, the recognition of channels and their properties is complicated because of the large bed elevation variations, such as shallow shoals and bed steps that render channels visually disconnected. We present and analyse two mathematically rigorous methods to identify channel networks from a terrain model of the river bed. Both methods construct a dense network of locally steepest-descent channels from saddle points on the terrain, and select a subset of channels with a certain minimum sediment volume between them. This is closely linked to the main mechanism of channel formation and change by displacement of sediment volume. The two methods differ in how they compute these sediment volumes: either globally through the entire length of the river, or locally. We compare the methods for the measured bathymetry of the Western Scheldt estuary, The Netherlands, over the past decades. The global method is overly sensitive to small changes elsewhere in the network compared to the local method. We conclude that the local method works best conceptually and for stability reasons. The associated concept of alluvial connectivity between channels in a network is thus the inverse of the volume of sediment that must be displaced to merge the channels. Our method opens up possibilities for new analyses as shown in two examples. First, it shows a clear pattern of scale dependence on volume of the total network length and of the number of nodes by a power law relation, showing that the smaller channels are relatively much shorter. Second, channel bifurcations were found to be predominantly mildly asymmetrical, which is unexpected from fluvial bifurcation theory.
－ BibTeX
```
@article{alluvial-connectivity-in-multi-channel-networks-in-rivers-and-estuaries:2022,
 title = {Alluvial Connectivity in Multi-Channel Networks in Rivers and Estuaries},
 author = {Willem Sonke and Maarten G. Kleinhans and Bettina Speckmann and Wout M. van Dijk and Matthew Hiatt},
 year = {2022},
 bookTitle = {Earth Surface Processes and Landforms},
}
```
[ PDF ]
Crossing Numbers of Beyond-Planar Graphs Revisited
Nathan van Beusekom, Irene Parada, and Bettina Speckmann.
Journal of Graph Algorithms and Applications, 26(1):149—170, 2022.

－ Abstract

Graph drawing beyond planarity focuses on drawings of high visual quality for non-planar graphs which are characterized by certain forbidden (edge) crossing configurations. A natural criterion for the quality of a drawing is the number of edge crossings. The question then arises whether beyond-planar drawings have a significantly larger crossing number than unrestricted drawings. Chimani et al. [GD'19] gave bounds for the ratio between the crossing number of three classes of beyond-planar graphs and the unrestricted crossing number. In this paper we extend their results to the main currently known classes of beyond-planar graphs characterized by forbidden edge configurations and answer several of their open questions.
－ BibTeX
```
@article{crossing-numbers-of-beyond-planar-graphs-revisited:2022,
    title = {Crossing Numbers of Beyond-Planar Graphs Revisited},
    author = {Nathan van Beusekom and Irene Parada and Bettina Speckmann},
    year = {2022},
    bookTitle = {Journal of Graph Algorithms and Applications},
}
```
[ PDF ]
Obstructing Classification Via Projection
Pantea Haghighatkhah, Wouter Meulemans, Bettina Speckmann, Jérôme Urhausen, and Kevin A.B. Verbeek.
Computing in Geometry and Topology, 1(1):2:1—2:21, 2022.

－ Abstract

Machine learning and data mining techniques are effective tools to classify large amounts of data. But they tend to preserve any inherent bias in the data, for example, with regards to gender or race. Removing such bias from data or the learned representations is quite challenging. In this paper we study a geometric problem which models a possible approach for bias removal. Our input is a set of points P in Euclidean space R^d and each point is labeled with k binary-valued properties. A priori we assume that it is `easy to classify the data according to each property. Our goal is to obstruct the classification according to one property by a suitable projection to a lower-dimensional Euclidean space R^m (m < d), while classification according to all other properties remains easy. What it means for classification to be easy depends on the classification model used. We first consider classification by linear separability as employed by support vector machines. We use Kirchberger's Theorem to show that, under certain conditions, a simple projection to R^(d-1) suffices to eliminate the linear separability of one of the properties whilst maintaining the linear separability of the other properties. We also study the problem of maximizing the linear ``inseparability of the chosen property. Second, we consider more complex forms of separability and prove a connection between the number of projections required to obstruct classification and the Helly-type properties of such separabilities.
－ BibTeX
```
@article{obstructing-classification-via-projection:2022,
 title = {Obstructing Classification Via Projection},
 author = {Pantea Haghighatkhah and Wouter Meulemans and Bettina Speckmann and Jérôme Urhausen and Kevin A.B. Verbeek},
 year = {2022},
 bookTitle = {Computing in Geometry and Topology},
}
```
[ PDF ]
Optimal Morphs of Planar Orthogonal Drawings
Arthur van Goethem, Bettina Speckmann, and Kevin Verbeek.
Journal of Computational Geometry, 13(1):263—297, 2022.

－ Abstract

We describe an algorithm that morphs between two planar orthogonal drawings ΓI and ΓO of a graph G, while preserving planarity and orthogonality. Necessarily drawings ΓI and ΓO must be equivalent, that is, there exists a homeomorphism of the plane that transforms ΓI into ΓO . Our morph uses a linear number of linear morphs (linear interpolations between two drawings) and preserves linear complexity throughout the process, thereby answering an open question from Biedl et al. (ACM Transactions on Algorithms, 2013). Our algorithm first unifies the two drawings to ensure an equal number of (virtual) bends on each edge. We then interpret bends as vertices which form obstacles for so-called wires: horizontal and vertical lines separating the vertices of ΓO . We can find corresponding wires in ΓI that share topological properties with the wires in ΓO . The structural difference between the two drawings can be captured by the spirality s of the wires in ΓI, which guides our morph from ΓI to ΓO . We prove that s = O(n) and that s + 1 linear morphs are always sufficient to morph between two planar orthogonal drawings, even for disconnected graphs.
－ BibTeX
```
@article{optimal-morphs-of-planar-orthogonal-drawings:2022,
 title = {Optimal Morphs of Planar Orthogonal Drawings},
 author = {Arthur van Goethem and Bettina Speckmann and Kevin Verbeek},
 year = {2022},
 bookTitle = {Journal of Computational Geometry},
}
```
[ PDF ]
Simultaneous Matrix Orderings for Graph Collections
Nathan van Beusekom, Wouter Meulemans, and Bettina Speckmann.
IEEE Transactions on Visualization and Computer Graphics, 28(1):1—10, 2022.

－ Abstract

Undirected graphs are frequently used to model phenomena that deal with interacting objects, such as social networks, brain activity and communication networks. The topology of an undirected graph G can be captured by an adjacency matrix; this matrix in turn can be visualized directly to give insight into the graph structure. Which visual patterns appear in such a matrix visualization crucially depends on the ordering of its rows and columns. Formally defining the quality of an ordering and then automatically computing a high-quality ordering are both challenging problems; however, effective heuristics exist and are used in practice. Often, graphs do not exist in isolation but as part of a collection of graphs on the same set of vertices, for example, brain scans over time or of different people. To visualize such graph collections, we need a single ordering that works well for all matrices simultaneously. The current state-of-the-art solves this problem by taking a (weighted) union over all graphs and applying existing heuristics. However, this union leads to a loss of information, specifically in those parts of the graphs which are different. We propose a collection-aware approach to avoid this loss of information and apply it to two popular heuristic methods: leaf order and barycenter. The de-facto standard computational quality metrics for matrix ordering capture only block-diagonal patterns (cliques). Instead, we propose to use Moran's I, a spatial auto-correlation metric, which captures the full range of established patterns. Moran's I refines previously proposed stress measures. Furthermore, the popular leaf order method heuristically optimizes a similar measure which further supports the use of Moran's I in this context. An ordering that maximizes Moran's I can be computed via solutions to the Traveling Salesperson Problem (TSP); orderings that approximate the optimal ordering can be computed more efficiently, using any of the approximation algorithms for metric TSP. We evaluated our methods for simultaneous orderings on real-world datasets using Moran's I as the quality metric. Our results show that our collection-aware approach matches or improves performance compared to the union approach, depending on the similarity of the graphs in the collection. Specifically, our Moran's I-based collection-aware leaf order implementation consistently outperforms other implementations. Our collection-aware implementations carry no significant additional computational costs.
－ BibTeX
```
@article{simultaneous-matrix-orderings-for-graph-collections:2022,
 title = {Simultaneous Matrix Orderings for Graph Collections},
 author = {Nathan van Beusekom and Wouter Meulemans and Bettina Speckmann},
 year = {2022},
 bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
}
```
Better Hit the Nail on the Head Than Beat Around the Bush: Removing Protected Attributes with a Single Projection.
Pantea Haghighatkhah, Antske Fokkens, Pia Sommerauer, Bettina Speckmann, and Kevin Verbeek.
Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing, pp. 8395—8416, 2022.

－ Abstract

Bias elimination and recent probing studies attempt to remove specific information from embedding spaces. Here it is important to remove as much of the target information as possible, while preserving any other information present. INLP is a popular recent method which removes specific information through iterative nullspace projections. Multiple iterations, however, increase the risk that information other than the target is negatively affected. We introduce two methods that find a single targeted projection: Mean Projection (MP, more efficient) and Tukey Median Projection (TMP, with theoretical guarantees). Our comparison between MP and INLP shows that (1) one MP projection removes linear separability based on the target and (2) MP has less impact on the overall space. Further analysis shows that applying random projections after MP leads to the same overall effects on the embedding space as the multiple projections of INLP. Applying one targeted (MP) projection hence is methodologically cleaner than applying multiple (INLP) projections that introduce random effects.
－ BibTeX
```
@article{better-hit-the-nail-on-the-head-than-beat-around-the-bush-removing-protected-attributes-with-a-single-projection-:2022,
 title = {Better Hit the Nail on the Head Than Beat Around the Bush: Removing Protected Attributes with a Single Projection.},
 author = {Pantea Haghighatkhah and Antske Fokkens and Pia Sommerauer and Bettina Speckmann and Kevin Verbeek},
 year = {2022},
 bookTitle = {Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing},
}
```
Brief Announcement: an Effective Geometric Communication Structure for Programmable Matter.
Irina Kostitsyna, Tom Peters, and Bettina Speckmann.
36th International Symposium on Distributed Computing (DISC 2022), pp. 47:1—47:3, 2022.

－ Abstract

The concept of programmable matter envisions a very large number of tiny and simple robot particles forming a smart material that can change its physical properties and shape based on the outcome of computation and movement performed by the individual particles in a concurrent manner. We use geometric insights to develop a new type of shortest path tree for programmable matter systems. Our feather trees utilize geometry to allow particles and information to traverse the programmable matter structure via shortest paths even in the presence of multiple overlapping trees.
－ BibTeX
```
@article{brief-announcement-an-effective-geometric-communication-structure-for-programmable-matter-:2022,
 title = {Brief Announcement: an Effective Geometric Communication Structure for Programmable Matter.},
 author = {Irina Kostitsyna and Tom Peters and Bettina Speckmann},
 year = {2022},
 bookTitle = {36th International Symposium on Distributed Computing (DISC 2022)},
}
```
Characterizing Uncertainty in the Visual Text Analysis Pipeline
Pantea Haghighatkhah, Bettina Speckmann, Mennatallah El-Assady, Jean-Daniel Fekete, Narges Mahyar, Carita Paradis, and Vasiliki Simaki.
2022 IEEE 7th Workshop on Visualization for the Digital Humanities (VIS4DH), pp. 25—30, 2022.

－ Abstract

Current visual text analysis approaches rely on sophisticated processing pipelines. Each step of such a pipeline potentially amplifies any uncertainties from the previous step. To ensure the comprehensibility and interoperability of the results, it is of paramount importance to clearly communicate the uncertainty not only of the output but also within the pipeline. In this paper, we characterize the sources of uncertainty along the visual text analysis pipeline. Within its three phases of labeling, modeling, and analysis, we identify six sources, discuss the type of uncertainty they create, and how they propagate. The goal of this paper is to bring the attention of the visualization community to additional types and sources of uncertainty in visual text analysis and to call for careful consideration, highlighting opportunities for future research.
－ BibTeX
```
@article{characterizing-uncertainty-in-the-visual-text-analysis-pipeline:2022,
    title = {Characterizing Uncertainty in the Visual Text Analysis Pipeline},
    author = {Pantea Haghighatkhah and Bettina Speckmann and Mennatallah  El-Assady and Jean-Daniel Fekete and Narges Mahyar and Carita Paradis and Vasiliki Simaki},
    year = {2022},
    bookTitle = {2022 IEEE 7th Workshop on Visualization for the Digital Humanities (VIS4DH)},
}
```
Compacting Squares: Input-Sensitive In-Place Reconfiguration of Sliding Squares
Hugo A. Akitaya, Erik D. Demaine, Matias Korman, Irina Kostitsyna, Irene Parada, Willem Sonke, Bettina Speckmann, Ryuhei Uehara, and Jules Wulms.
18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022), pp. 4:1—4:19, 2022.

－ Abstract

Edge-connected configurations of square modules, which can reconfigure through so-called sliding moves, are a well-established theoretical model for modular robots in two dimensions. Dumitrescu and Pach [Graphs and Combinatorics, 2006] proved that it is always possible to reconfigure one edge-connected configuration of n squares into any other using at most O(n2) sliding moves, while keeping the configuration connected at all times. For certain pairs of configurations, reconfiguration may require ω(n2) sliding moves. However, significantly fewer moves may be sufficient. We prove that it is NP-hard to minimize the number of sliding moves for a given pair of edge-connected configurations. On the positive side we present Gather&Compact, an input-sensitive in-place algorithm that requires only O( Pn) sliding moves to transform one configuration into the other, where P is the maximum perimeter of the two bounding boxes. The squares move within the bounding boxes only, with the exception of at most one square at a time which may move through the positions adjacent to the bounding boxes. The O( Pn) bound never exceeds O(n2), and is optimal (up to constant factors) among all bounds parameterized by just n and P. Our algorithm is built on the basic principle that well-connected components of modular robots can be transformed efficiently. Hence we iteratively increase the connectivity within a configuration, to finally arrive at a single solid xy-monotone component. We implemented Gather&Compact and compared it experimentally to the in-place modification by Moreno and Sacristán [EuroCG 2020] of the Dumitrescu and Pach algorithm (MSDP). Our experiments show that Gather&Compact consistently outperforms MSDP by a significant margin, on all types of square configurations.
－ BibTeX
```
@article{compacting-squares-input-sensitive-in-place-reconfiguration-of-sliding-squares:2022,
 title = {Compacting Squares: Input-Sensitive In-Place Reconfiguration of Sliding Squares},
 author = {Hugo A. Akitaya and Erik D. Demaine and Matias Korman and Irina Kostitsyna and Irene Parada and Willem Sonke and Bettina Speckmann and Ryuhei Uehara and Jules Wulms},
 year = {2022},
 bookTitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
}
```

Physically Consistent Map Matching
Bram A. Custers, Wouter Meulemans, Marcel J.M. Roeloffzen, Bettina Speckmann, and Kevin A.B. Verbeek.
Physically Consistent Map Matching, 2022.

－ BibTeX

@article{physically-consistent-map-matching:2022,
    title = {Physically Consistent Map Matching},
    author = {Bram A. Custers and Wouter Meulemans and Marcel J.M. Roeloffzen and Bettina Speckmann and Kevin A.B. Verbeek},
    year = {2022},
    bookTitle = {Physically Consistent Map Matching},
}

[ PDF ]

Preprocessing Imprecise Points for the Pareto Front.
Ivor van der Hoog, Irina Kostitsyna, Maarten Löffler, and Bettina Speckmann.
ACM-SIAM Symposium on Discrete Algorithms (SODA22), pp. 3144—3167, 2022.

－ Abstract

The preprocessing model for uncertain data models geometric imprecision of algorithmic input and provides a framework for working with it. In this model, we are given a set of regions ℛ which model the uncertainty associated with an unknown set of points P. There are two phases: a preprocessing phase, in which we have access only to ℛ, followed by a reconstruction phase, in which we have access to points in P, possibly at a certain retrieval cost C per point. For a given algorithmic problem, the goal in this model is to perform as much of the necessary computations as possible in the preprocessing phase, so that the amount of time spent in the reconstruction phase is minimized. In this paper, we investigate the following algorithmic question: how fast can we compute the Pareto front of P in the preprocessing model? We show that if ℛ is a set of pairwise-disjoint axis-aligned rectangles then we can preprocess ℛ to reconstruct the Pareto front of P efficiently. In contrast to earlier work in the preprocessing model, our solution achieves sublinear reconstruction time when the output complexity is sublinear. To refine our algorithmic analysis, we introduce a new notion of algorithmic optimality which relates to the entropy of the uncertainty regions. Our proposed uncertainty-region optimality falls on the spectrum between worst-case optimality and instance optimality. Our results are worst-case optimal, but we prove that instance optimality is unobtainable for a wide class of problems in the preprocessing model.
－ BibTeX
```
@article{preprocessing-imprecise-points-for-the-pareto-front-:2022,
 title = {Preprocessing Imprecise Points for the Pareto Front.},
 author = {Ivor van der Hoog and Irina Kostitsyna and Maarten Löffler and Bettina Speckmann},
 year = {2022},
 bookTitle = {ACM-SIAM Symposium on Discrete Algorithms (SODA22)},
}
```
Story Trees
Pantea Haghighatkhah, Antske Fokkens, Pia Sommerauer, Bettina Speckmann, and Kevin Verbeek.
2022 Language Resources and Evaluation Conference, LREC 2022, pp. 2413—2429, 2022.

－ Abstract

Topological Data Analysis (TDA) focuses on the inherent shape of (spatial) data. As such, it may provide useful methods to explore spatial representations of linguistic data (embeddings) which have become central in NLP. In this paper we aim to introduce TDA to researchers in language technology. We use TDA to represent document structure as so-called story trees. Story trees are hierarchical representations created from semantic vector representations of sentences via persistent homology. They can be used to identify and clearly visualize prominent components of a story line. We showcase their potential by using story trees to create extractive summaries for news stories.
－ BibTeX
```
@article{story-trees:2022,
 title = {Story Trees},
 author = {Pantea Haghighatkhah and Antske Fokkens and Pia Sommerauer and Bettina Speckmann and Kevin Verbeek},
 year = {2022},
 bookTitle = {2022 Language Resources and Evaluation Conference, LREC 2022},
}
```

Fast Reconfiguration for Programmable Matter
Irina Kostitsyna, Tom Peters, and Bettina Speckmann.
pp. 365—371, 2022.

－ BibTeX

@article{fast-reconfiguration-for-programmable-matter:2022,
    title = {Fast Reconfiguration for Programmable Matter},
    author = {Irina Kostitsyna and Tom Peters and Bettina Speckmann},
    year = {2022},
    bookTitle = {},
}

[ PDF ]

Kinetic Group Density in 1D
Kevin A. Buchin, Max J.M. van Mulken, Bettina Speckmann, and Kevin A.B. Verbeek.
pp. 47:1—47:6, 2022.

－ Abstract

We are interested in tracking the overall structure of a single group of entities (e.g., people, wildlife, or vehicles) over time. As a first step, we investigate how to characterize and kinetically maintain the density of a one-dimensional group, represented by a point set P of size n. We achieve this with a kinetic data structure that maintains the critical points of an estimate of the density function of P, which is obtained using kernel density estimation. Our KDS is local, compact, responsive, and weakly efficient. However, the total number of events can be Θ(n^2). Therefore, we also show how to maintain an ɛ-approximation of the critical points using a coreset of the trajectories of the points in P. The coreset has size O(1/ɛ^2 log(1/ɛ)), for ɛ > 0, and approximates the critical points well in a topological sense.
－ BibTeX
```
@article{kinetic-group-density-in-1d:2022,
    title = {Kinetic Group Density in 1D},
    author = {Kevin A. Buchin and Max J.M. van Mulken and Bettina Speckmann and Kevin A.B. Verbeek},
    year = {2022},
    bookTitle = {},
}
```

2021

A Simple Pipeline for Coherent Grid Maps
Wouter Meulemans, Max F.M. Sondag, and Bettina Speckmann.
IEEE Transactions on Visualization and Computer Graphics, 27(2):1236—1246, 2021.

－ Abstract

Grid maps are spatial arrangements of simple tiles (often squares or hexagons), each of which represents a spatial element. They are an established, effective way to show complex data per spatial element, using visual encodings within each tile ranging from simple coloring to nested small-multiples visualizations. An effective grid map is coherent with the underlying geographic space: the tiles maintain the contiguity, neighborhoods and identifiability of the corresponding spatial elements, while the grid map as a whole maintains the global shape of the input. Of particular importance are salient local features of the global shape which need to be represented by tiles assigned to the appropriate spatial elements. State-of-the-art techniques can adequately deal only with simple cases, such as close-to-uniform spatial distributions or global shapes that have few characteristic features. We introduce a simple fully-automated 3-step pipeline for computing coherent grid maps. Each step is a well-studied problem: shape decomposition based on salient features, tile-based Mosaic Cartograms, and point-set matching. Our pipeline is a seamless composition of existing techniques for these problems and results in high-quality grid maps. We provide an implementation, demonstrate the efficacy of our approach on various complex datasets, and compare it to the state-of-the-art.
－ BibTeX
```
@article{a-simple-pipeline-for-coherent-grid-maps:2021,
    title = {A Simple Pipeline for Coherent Grid Maps},
    author = {Wouter Meulemans and Max F.M. Sondag and Bettina Speckmann},
    year = {2021},
    bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
}
```
[ PDF ]
[ PDF ]
Maximum Physically Consistent Trajectories
Bram A. Custers, Frank Staals, Bettina Speckmann, Wouter Meulemans, and Mees van de Kerkhof.
ACM Transactions on Spatial Algorithms and Systems , 7(4):, 2021.

－ Abstract

Trajectories are usually collected with physical sensors, which are prone to errors and cause outliers in the data. We aim to identify such outliers via the physical properties of the tracked entity, that is, we consider its physical possibility to visit combinations of measurements. We describe optimal algorithms to compute maximum subsequences of measurements that are consistent with (simplified) physics models. Our results are output-sensitive with respect to the number k of outliers in a trajectory of n measurements. Specifically, we describe an O(n log n log2 k)-time algorithm for 2D trajectories using a model with unbounded acceleration but bounded velocity, and an O(nk)-time algorithm for any model where consistency is “concatenable”: a consistent subsequence that ends where another begins together form a consistent sequence. We also consider acceleration-bounded models that are not concatenable. We show how to compute the maximum subsequence for such models in O(n k2 log k) time, under appropriate realism conditions. Finally, we experimentally explore the performance of our algorithms on several large real-world sets of trajectories. Our experiments show that we are generally able to retain larger fractions of noisy trajectories than previous work and simpler greedy approaches. We also observe that the speed-bounded model may in practice approximate the acceleration-bounded model quite well, though we observed some variation between datasets.
－ BibTeX
```
@article{maximum-physically-consistent-trajectories:2021,
    title = {Maximum Physically Consistent Trajectories},
    author = {Bram A. Custers and Frank Staals and Bettina Speckmann and Wouter Meulemans and Mees van de Kerkhof},
    year = {2021},
    bookTitle = {ACM Transactions on Spatial Algorithms and Systems },
}
```
[ PDF ]
The Vulnerability of Tidal Flats and Multi-Channel Estuaries to Dredging and Disposal
Wout van Dijk, Jana Cox, Jasper Leuven, Jelmer Cleveringa, Marcel Taal, Matthew Hiatt, Willem M. Sonke, Kevin A.B. Verbeek, Bettina Speckmann, and Maarten G. Kleinhans.
Anthropocene Coasts, 4:36—60, 2021.

－ Abstract

Shipping fairways in estuaries are continuously dredged to maintain access for large vessels to major ports. However, several estuaries worldwide show adverse side effects to dredging activities, in particular affecting morphology and ecologically valuable habitats. We used physical scale experiments, field assessments of the Western Scheldt estuary (the Netherlands), and morphodynamic model runs to analyse the effects of dredging and future stresses (climate and sediment management) on a multi-channel system and its ecologically valuable intertidal flats. All methods indicate that dredging and disposal strategies are unfavourable to long-term morphology because dredging creates and propagates the imbalance between shallow and deeper parts of the estuary, causing a loss of valuable connecting channels and fixation of the tidal flats and main channel positions, while countering adverse effects by disposal strategy has limited effectiveness. Changing the disposal strategy towards main channel scour disposal can be economically and ecologically beneficial for the preservation of the multi-channel system. Further channel deepening will accelerate the adverse side effects, whereas future sea-level rise may revive the multi-channel system.
－ BibTeX
```
@article{the-vulnerability-of-tidal-flats-and-multi-channel-estuaries-to-dredging-and-disposal:2021,
    title = {The Vulnerability of Tidal Flats and Multi-Channel Estuaries to Dredging and Disposal},
    author = {Wout van Dijk and Jana Cox and Jasper  Leuven and Jelmer Cleveringa and Marcel Taal and Matthew Hiatt and Willem M. Sonke and Kevin A.B. Verbeek and Bettina Speckmann and Maarten G. Kleinhans},
    year = {2021},
    bookTitle = {Anthropocene Coasts},
}
```
[ PDF ]
Coordinated Schematization for Visualizing Mobility Patterns on Networks
Bram A. Custers, Wouter Meulemans, Bettina Speckmann, and Kevin Verbeek.
11th International Conference on Geographic Information Science - Part II, GIScience 2021, pp. VII, 2021.

－ Abstract

GPS trajectories of vehicles moving on a road network are a valuable source of traffic information. However, the sheer volume of available data makes it challenging to identify and visualize salient patterns. Meaningful visual summaries of trajectory collections require that both the trajectories and the underlying network are aggregated and simplified in a coherent manner. In this paper we propose a coordinated fully-automated pipeline for computing a schematic overview of mobility patterns from a collection of trajectories on a street network. Our pipeline utilizes well-known building blocks from GIS, automated cartography, and trajectory analysis: map matching, road selection, schematization, movement patterns, and metro-map style rendering. We showcase the results of our pipeline on two real-world trajectory collections around The Hague and Beijing.
－ BibTeX
```
@article{coordinated-schematization-for-visualizing-mobility-patterns-on-networks:2021,
 title = {Coordinated Schematization for Visualizing Mobility Patterns on Networks},
 author = {Bram A. Custers and Wouter Meulemans and Bettina Speckmann and Kevin Verbeek},
 year = {2021},
 bookTitle = {11th International Conference on Geographic Information Science - Part II, GIScience 2021},
}
```
Harmonious Simplification of Isolines
Arthur van Goethem, Wouter Meulemans, Andreas W. Reimer, and Bettina Speckmann.
Proceedings of the 11th International Conference on Geographic Information Science, pp. 8:1—8:16, 2021.

－ Abstract

Current techniques for simplification focus on reducing complexity while maintaining the geometric similarity to the input. For isolines that jointly describe a scalar field, however, we postulate that geometric similarity of each isoline separately is not sufficient. Rather, we need to maintain the harmony between these isolines to make them visually relate and describe the structures of the underlying terrain. Based on principles of manual cartography, we propose an algorithm for simplifying isolines while considering harmony explicitly. Our preliminary visual and quantitative results suggest that our algorithm is effective.
－ BibTeX
```
@article{harmonious-simplification-of-isolines:2021,
 title = {Harmonious Simplification of Isolines},
 author = {Arthur van Goethem and Wouter Meulemans and Andreas W. Reimer and Bettina Speckmann},
 year = {2021},
 bookTitle = {Proceedings of the 11th International Conference on Geographic Information Science},
}
```
Obstructing Classification Via Projection
Pantea Haghighatkhah, Wouter Meulemans, Bettina Speckmann, Jérôme Urhausen, and Kevin Verbeek.
46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021, 2021.

－ Abstract

Machine learning and data mining techniques are effective tools to classify large amounts of data. But they tend to preserve any inherent bias in the data, for example, with regards to gender or race. Removing such bias from data or the learned representations is quite challenging. In this paper we study a geometric problem which models a possible approach for bias removal. Our input is a set of points P in Euclidean space Rd and each point is labeled with k binary-valued properties. A priori we assume that it is "easy"to classify the data according to each property. Our goal is to obstruct the classification according to one property by a suitable projection to a lower-dimensional Euclidean space Rm (m < d), while classification according to all other properties remains easy. What it means for classification to be easy depends on the classification model used. We first consider classification by linear separability as employed by support vector machines. We use Kirchberger's Theorem to show that, under certain conditions, a simple projection to Rd1 suffices to eliminate the linear separability of one of the properties whilst maintaining the linear separability of the other properties. We also study the problem of maximizing the linear "inseparability"of the chosen property. Second, we consider more complex forms of separability and prove a connection between the number of projections required to obstruct classification and the Helly-type properties of such separabilities. 
－ BibTeX
```
@article{obstructing-classification-via-projection:2021,
 title = {Obstructing Classification Via Projection},
 author = {Pantea Haghighatkhah and Wouter Meulemans and Bettina Speckmann and Jérôme Urhausen and Kevin Verbeek},
 year = {2021},
 bookTitle = {46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021},
}
```
Polygon-Universal Graphs
Tim Ophelders, Ignaz Rutter, Bettina Speckmann, and Kevin Verbeek.
37th International Symposium on Computational Geometry, SoCG 2021, 2021.

－ Abstract

We study a fundamental question from graph drawing: given a pair (G,C) of a graph G and a cycle C in G together with a simple polygon P, is there a straight-line drawing of G inside P which maps C to P? We say that such a drawing of (G,C) respects P. We fully characterize those instances (G,C) which are polygon-universal, that is, they have a drawing that respects P for any simple (not necessarily convex) polygon P. Specifically, we identify two necessary conditions for an instance to be polygon-universal. Both conditions are based purely on graph and cycle distances and are easy to check. We show that these two conditions are also sufficient. Furthermore, if an instance (G,C) is planar, that is, if there exists a planar drawing of G with C on the outer face, we show that the same conditions guarantee for every simple polygon P the existence of a planar drawing of (G,C) that respects P. If (G,C) is polygon-universal, then our proofs directly imply a linear-time algorithm to construct a drawing that respects a given polygon P.
－ BibTeX
```
@article{polygon-universal-graphs:2021,
 title = {Polygon-Universal Graphs},
 author = {Tim Ophelders and Ignaz Rutter and Bettina Speckmann and Kevin Verbeek},
 year = {2021},
 bookTitle = {37th International Symposium on Computational Geometry, SoCG 2021},
}
```
Route Reconstruction from Traffic Flow Via Representative Trajectories
Bram A. Custers, Kevin A.B. Verbeek, Wouter Meulemans, and Bettina Speckmann.
Proc. 29th International Conference on Advances in Geographic Information (SIGSPATIAL), pp. 41—52, 2021.

－ Abstract

Understanding human mobility patterns is an important aspect of traffic analysis and urban planning. Trajectory data provide detailed views on specific routes, but typically do not capture all traffic. On the other hand, loop detectors built into the road network capture all traffic flow at specific locations, but provide no information on the individual routes. Given a set of loop-detector measurements as well as a (small) set of representative trajectories, our goal is to investigate how one can effectively combine these two partial data sources to create a more complete picture of the underlying mobility patterns. Specifically, we want to reconstruct a realistic set of routes from the loop-detector data, using the given trajectories as representatives of typical behavior. We model the loop-detector data as a network flow field that needs to be covered by the reconstructed routes and we capture the realism of the routes via the strong Fréchet distance to the representative trajectories. We prove that several forms of the resulting algorithmic problem are NP-hard. Hence we explore heuristic approaches which decompose the flow well while following the representative trajectories to varying degrees. We propose an iterative Fréchet Routes (FR) heuristic which generates candidates routes with bounded Fréchet distance to the representative trajectories. We also describe a variant of multi-commodity min-cost flow (MCMCF) which is only loosely coupled to the trajectories. We perform an extensive experimental evaluation of our two proposed approaches in comparison to a global min-cost flow (GMCF), which is essentially agnostic to the representative trajectories. To make meaningful claims in terms of quality, we derive a ground truth by map-matching real-world trajectories. We find that GMCF explains the flow best, but produces a large number of often nonsensical routes (significantly more than the ground truth). MCMCF produces a large number of mostly realistic routes which explain the flow reasonably well. In contrast, FR produces much smaller sets of realistic routes which still explain the flow well, at the cost of a higher running time. Finally, we report on the results of a case study which combines real-world loop detector data and representative trajectories for the region around The Hague, the Netherlands. 
－ BibTeX
```
@article{route-reconstruction-from-traffic-flow-via-representative-trajectories:2021,
 title = {Route Reconstruction from Traffic Flow Via Representative Trajectories},
 author = {Bram A. Custers and Kevin A.B. Verbeek and Wouter Meulemans and Bettina Speckmann},
 year = {2021},
 bookTitle = {Proc. 29th International Conference on Advances in Geographic Information (SIGSPATIAL)},
}
```
[ PDF ]
Stable Visual Summaries for Trajectory Collections
Jules Wulms, Juri Buchmuller, Wouter Meulemans, Kevin Verbeek, and Bettina Speckmann.
Proceedings - 2021 IEEE 14th Pacific Visualization Symposium, PacificVis 2021, pp. 61—70, 2021.

－ Abstract

The availability of devices that track moving objects has led to an explosive growth in trajectory data. When exploring the resulting large trajectory collections, visual summaries are a useful tool to identify time intervals of interest. A typical approach is to represent the spatial positions of the tracked objects at each time step via a one-dimensional ordering; visualizations of such orderings can then be placed in temporal order along a time line. There are two main criteria to assess the quality of the resulting visual summary: spatial quality - how well does the ordering capture the structure of the data at each time step, and stability - how coherent are the orderings over consecutive time steps or temporal ranges?In this paper we introduce a new Stable Principal Component (SPC) method to compute such orderings, which is explicitly parameterized for stability, allowing a trade-off between the spatial quality and stability. We conduct extensive computational experiments that quantitatively compare the orderings produced by ours and other stable dimensionality-reduction methods to various state-of-the-art approaches using a set of well-established quality metrics that capture spatial quality and stability. We conclude that stable dimensionality reduction outperforms existing methods on stability, without sacrificing spatial quality or efficiency; in particular, our new SPC method does so at a fraction of the computational costs.
－ BibTeX
```
@article{stable-visual-summaries-for-trajectory-collections:2021,
 title = {Stable Visual Summaries for Trajectory Collections},
 author = {Jules Wulms and Juri Buchmuller and Wouter Meulemans and Kevin Verbeek and Bettina Speckmann},
 year = {2021},
 bookTitle = {Proceedings - 2021 IEEE 14th Pacific Visualization Symposium, PacificVis 2021},
}
```

Coordinating Programmable Matter Via Shortest Path Trees
Irina Kostitsyna, Tom Peters, and Bettina Speckmann.
pp. 234—240, 2021.

－ BibTeX

@article{coordinating-programmable-matter-via-shortest-path-trees:2021,
    title = {Coordinating Programmable Matter Via Shortest Path Trees},
    author = {Irina Kostitsyna and Tom Peters and Bettina Speckmann},
    year = {2021},
    bookTitle = {},
}

[ PDF ]

Crossing Numbers of Beyond-Planar Graphs Revisited
Nathan van Beusekom, Irene Parada, and Bettina Speckmann.
pp. 36:1—36:7, 2021.

－ Abstract

Graph drawing beyond planarity focuses on drawings of high visual quality for non-planar graphs which are characterized by certain forbidden edge configurations. A natural criterion for the quality of a drawing is the number of edge crossings. The question then arises whether beyond-planar drawings have a significantly larger crossing number than unrestricted drawings. Chimani et al. [GD’19] gave bounds for the ratio between the crossing number of three classes of beyond-planar graphs and the unrestricted crossing number. In this paper we extend their results to the main currently known classes of beyond-planar graphs characterized by forbidden edge configurations and answer several of their open questions.
－ BibTeX
```
@article{crossing-numbers-of-beyond-planar-graphs-revisited:2021,
 title = {Crossing Numbers of Beyond-Planar Graphs Revisited},
 author = {Nathan van Beusekom and Irene Parada and Bettina Speckmann},
 year = {2021},
 bookTitle = {},
}
```
[ PDF ]
Near-Delaunay Metrics
Nathan van Beusekom, Kevin A. Buchin, Hidde O. Koerts, Wouter Meulemans, Benjamin Rodatz, and Bettina Speckmann.
pp. 1—11, 2021.

－ Abstract

We study metrics that assess how close a triangulation is to being a Delaunay triangulation, for use in contexts where a good triangulation is desired but constraints (e.g., maximum degree) prevent the use of the Delaunay triangulation itself. Our near-Delaunay metrics derive from common Delaunay properties and satisfy a basic set of design criteria, such as being invariant under similarity transformations. We compare the metrics, showing that each can make different judgments as to which triangulation is closer to Delaunay. We also present a preliminary experiment, showing how optimizing for these metrics under different constraints gives similar, but not necessarily identical results, on random and constructed small point sets.
－ BibTeX
```
@article{near-delaunay-metrics:2021,
    title = {Near-Delaunay Metrics},
    author = {Nathan van Beusekom and Kevin A. Buchin and Hidde O. Koerts and Wouter Meulemans and Benjamin Rodatz and Bettina Speckmann},
    year = {2021},
    bookTitle = {},
}
```
[ PDF ]

Route Reconstruction from Traffic Flow Via Representative Trajectories
Bram A. Custers, Bettina Speckmann, Wouter Meulemans, and Kevin A.B. Verbeek.
2021.

－ BibTeX

@article{route-reconstruction-from-traffic-flow-via-representative-trajectories:2021,
    title = {Route Reconstruction from Traffic Flow Via Representative Trajectories},
    author = {Bram A. Custers and Bettina Speckmann and Wouter Meulemans and Kevin A.B. Verbeek},
    year = {2021},
    bookTitle = {},
}

2020

Can Real Social Epistemic Networks Deliver the Wisdom of Crowds?
Emily Sullivan, Max Sondag, Ignaz Rutter, Wouter Meulemans, Scott Cunningham, Bettina Speckmann, and Mark Alfano.
Oxford Studies in Experimental Philosophy, Volume 3, pp. 29—63, 2020.

－ BibTeX

@article{can-real-social-epistemic-networks-deliver-the-wisdom-of-crowds-:2020,
    title = {Can Real Social Epistemic Networks Deliver the Wisdom of Crowds?},
    author = {Emily Sullivan and Max Sondag and Ignaz Rutter and Wouter Meulemans and Scott Cunningham and Bettina Speckmann and Mark Alfano},
    year = {2020},
    bookTitle = {Oxford Studies in Experimental Philosophy, Volume 3},
}

Geometry and Topology of Estuary and Braided River Channel Networks Automatically Extracted from Topographic Data
Matthew Hiatt, Willem M. Sonke, Elisabeth Addink, Wout van Dijk, Marc J. van Kreveld, Tim A.E. Ophelders, Kevin A.B. Verbeek, Joyce Vlaming, Bettina Speckmann, and Maarten G. Kleinhans.
Journal of Geophysical Research: Earth Surface, 125(1):, 2020.

－ Abstract

Automatic extraction of channel networks from topography in systems with multiple interconnected channels, like braided rivers and estuaries, remains a major challenge in hydrology and geomorphology. Representing channelized systems as networks provides a mathematical framework for analyzing transport and geomorphology. In this paper, we introduce a mathematically rigorous methodology and software for extracting channel network topology and geometry from digital elevation models (DEMs) and analyze such channel networks in estuaries and braided rivers. Channels are represented as network links, while channel confluences and bifurcations are represented as network nodes. We analyze and compare DEMs from the field and those generated by numerical modeling. We use a metric called the volume parameter that characterizes the volume of deposited material separating channels to quantify the volume of reworkable sediment deposited between links, which is a measure for the spatial scale associated with each network link. Scale asymmetry is observed in most links downstream of bifurcations, indicating geometric asymmetry and bifurcation stability. The length of links relative to system size scales with volume parameter value to the power of 0.24–0.35, while the number of links decreases and does not exhibit power law behavior. Link depth distributions indicate that the estuaries studied tend to organize around a deep main channel that exists at the largest scale while braided rivers have channel depths that are more evenly distributed across scales. The methods and results presented establish a benchmark for quantifying the topology and geometry of multichannel networks from DEMs with a new automatic extraction tool.
－ BibTeX
```
@article{geometry-and-topology-of-estuary-and-braided-river-channel-networks-automatically-extracted-from-topographic-data:2020,
    title = {Geometry and Topology of Estuary and Braided River Channel Networks Automatically Extracted from Topographic Data},
    author = {Matthew Hiatt and Willem M. Sonke and Elisabeth Addink and Wout van Dijk and Marc J. van Kreveld and Tim A.E. Ophelders and Kevin A.B. Verbeek and Joyce Vlaming and Bettina Speckmann and Maarten G. Kleinhans},
    year = {2020},
    bookTitle = {Journal of Geophysical Research: Earth Surface},
}
```
[ PDF ]
Quantitative Comparison of Time-Dependent Treemaps
E. Vernier, M. Sondag, J. Comba, B. Speckmann, A. Telea, and K. Verbeek.
Computer Graphics Forum, 39(3):393—404, 2020.

－ Abstract

Rectangular treemaps are often the method of choice to visualize large hierarchical datasets. Nowadays such datasets are available over time, hence there is a need for (a) treemaps that can handle time-dependent data, and (b) corresponding quality criteria that cover both a treemap's visual quality and its stability over time. In recent years a wide variety of (stable) treemapping algorithms has been proposed, with various advantages and limitations. We aim to provide insights to researchers and practitioners to allow them to make an informed choice when selecting a treemapping algorithm for specific applications and data. To this end, we perform an extensive quantitative evaluation of rectangular treemaps for time-dependent data. As part of this evaluation we propose a novel classification scheme for time-dependent datasets. Specifically, we observe that the performance of treemapping algorithms depends on the characteristics of the datasets used. We identify four potential representative features that characterize time-dependent hierarchical datasets and classify all datasets used in our experiments accordingly. We experimentally test the validity of this classification on more than 2000 datasets, and analyze the relative performance of 14 state-of-the-art rectangular treemapping algorithms across varying features. Finally, we visually summarize our results with respect to both visual quality and stability to aid users in making an informed choice among treemapping algorithms. All datasets, metrics, and algorithms are openly available to facilitate reuse and further comparative studies.
－ BibTeX
```
@article{quantitative-comparison-of-time-dependent-treemaps:2020,
 title = {Quantitative Comparison of Time-Dependent Treemaps},
 author = {E. Vernier and M. Sondag and J. Comba and B. Speckmann and A. Telea and K. Verbeek},
 year = {2020},
 bookTitle = {Computer Graphics Forum},
}
```
Vulnerability in Social Epistemic Networks
Emily Sullivan, Max Sondag, Ignaz Rutter, Wouter Meulemans, Scott Cunningham, Bettina Speckmann, and Mark Alfano.
International Journal of Philosophical Studies, 28(5):731—753, 2020.

－ Abstract

Social epistemologists should be well-equipped to explain and evaluate the growing vulnerabilities associated with filter bubbles, echo chambers, and group polarization in social media. However, almost all social epistemology has been built for social contexts that involve merely a speaker-hearer dyad. Filter bubbles, echo chambers, and group polarization all presuppose much larger and more complex network structures. In this paper, we lay the groundwork for a properly social epistemology that gives the role and structure of networks their due. In particular, we formally define epistemic constructs that quantify the structural epistemic position of each node within an interconnected network. We argue for the epistemic value of a structure that we call the (m,k)-observer. We then present empirical evidence that (m,k)-observers are rare in social media discussions of controversial topics, which suggests that people suffer from serious problems of epistemic vulnerability. We conclude by arguing that social epistemologists and computer scientists should work together to develop minimal interventions that improve the structure of epistemic networks.
－ BibTeX
```
@article{vulnerability-in-social-epistemic-networks:2020,
 title = {Vulnerability in Social Epistemic Networks},
 author = {Emily Sullivan and Max Sondag and Ignaz Rutter and Wouter Meulemans and Scott Cunningham and Bettina Speckmann and Mark Alfano},
 year = {2020},
 bookTitle = {International Journal of Philosophical Studies},
}
```
[ PDF ]
Hiding Sliding Cubes - Why Reconfiguring Modular Robots is Not Easy
Tillmann Miltzow, Irene Parada, Willem Sonke, Bettina Speckmann, and Jules Wulms.
36th International Symposium on Computational Geometry (SoCG 2020), pp. 78:1—78:5, 2020.

－ Abstract

Face-connected configurations of cubes are a common model for modular robots in three dimensions. In this abstract and the accompanying video we study reconfigurations of such modular robots using so-called sliding moves. Using sliding moves, it is always possible to reconfigure one face-connected configuration of n cubes into any other, while keeping the robot connected at all stages of the reconfiguration. For certain configurations Ω(n 2) sliding moves are necessary. In contrast, the best current upper bound is O(n 3). It has been conjectured that there is always a cube on the outside of any face-connected configuration of cubes which can be moved without breaking connectivity. The existence of such a cube would immediately imply a straight-forward O(n 2) reconfiguration algorithm. However, we present a configuration of cubes such that no cube on the outside can move without breaking connectivity. In other words, we show that this particular avenue towards an O(n 2) reconfiguration algorithm for face-connected cubes is blocked. 
－ BibTeX
```
@article{hiding-sliding-cubes-why-reconfiguring-modular-robots-is-not-easy:2020,
 title = {Hiding Sliding Cubes - Why Reconfiguring Modular Robots is Not Easy},
 author = {Tillmann Miltzow and Irene Parada and Willem Sonke and Bettina Speckmann and Jules Wulms},
 year = {2020},
 bookTitle = {36th International Symposium on Computational Geometry (SoCG 2020)},
}
```
Ordered Strip Packing
K. Buchin, D. Kosolobov, W. Sonke, B. Speckmann, and K. Verbeek.
LATIN 2020, pp. 258—270, 2020.

－ Abstract

We study an ordered variant of the well-known strip packing problem, which is motivated by applications in visualization and typography. Our input consists of a maximum width W and an ordered list of n blocks (rectangles). The goal is to pack the blocks into rows (not exceeding W) while obeying the given order and minimizing either the number of rows or the total height of the drawing. We consider two variants: (1) non-overlapping row drawing (NORD), where distinct rows cannot share y-coordinates, and (2) overlapping row drawing (ORD), where consecutive rows may overlap vertically. We present an algorithm that computes the minimum-height NORD in O(n) time. Further, we study the worst-case tradeoffs between the two optimization criteria—number of rows and total height—for both NORD and ORD. Surprisingly, we show that the minimum-height ORD may require Ω(log n/ log log n) times as many rows as the minimum-row ORD. The proof of the matching upper bound employs a novel application of information entropy.
－ BibTeX
```
@article{ordered-strip-packing:2020,
 title = {Ordered Strip Packing},
 author = {K. Buchin and D. Kosolobov and W. Sonke and B. Speckmann and K. Verbeek},
 year = {2020},
 bookTitle = {LATIN 2020},
}
```
[ PDF ]

Simplification with Parallelism
Arthur I. van Goethem, Wouter Meulemans, Andreas Reimer, and Bettina Speckmann.
Abstr. 23rd ICA Workshop on Generalisation and Multiple Representation, 2020.

－ BibTeX

@article{simplification-with-parallelism:2020,
    title = {Simplification with Parallelism},
    author = {Arthur I. van Goethem and Wouter Meulemans and Andreas Reimer and Bettina Speckmann},
    year = {2020},
    bookTitle = {Abstr. 23rd ICA Workshop on Generalisation and Multiple Representation},
}

Uncertainty Treemaps
Max Sondag, Wouter Meulemans, Christoph Schulz, Kevin Verbeek, Daniel Weiskopf, and Bettina Speckmann.
2020 IEEE Pacific Visualization Symposium, PacificVis 2020 - Proceedings, pp. 111—120, 2020.

－ Abstract

Rectangular treemaps visualize hierarchical numerical data by recursively partitioning an input rectangle into smaller rectangles whose areas match the data. Numerical data often has uncertainty associated with it. To visualize uncertainty in a rectangular treemap, we identify two conflicting key requirements: (i) to assess the data value of a node in the hierarchy, the area of its rectangle should directly match its data value, and (ii) to facilitate comparison between data and uncertainty, uncertainty should be encoded using the same visual variable as the data, that is, area. We present Uncertainty Treemaps, which meet both requirements simultaneously by introducing the concept of hierarchical uncertainty masks. First, we define a new cost function that measures the quality of Uncertainty Treemaps. Then, we show how to adapt existing treemapping algorithms to support uncertainty masks. Finally, we demonstrate the usefulness and quality of our technique through an expert review and a computational experiment on real-world datasets.
－ BibTeX
```
@article{uncertainty-treemaps:2020,
 title = {Uncertainty Treemaps},
 author = {Max Sondag and Wouter Meulemans and Christoph Schulz and Kevin Verbeek and Daniel Weiskopf and Bettina Speckmann},
 year = {2020},
 bookTitle = {2020 IEEE Pacific Visualization Symposium, PacificVis 2020 - Proceedings},
}
```
[ PDF ]
Volume from Outlines on Terrains
Marc Van Kreveld, Tim Ophelders, Willem Sonke, Bettina Speckmann, and Kevin Verbeek.
11th International Conference on Geographic Information Science, GIScience 2021, pp. 16:1—16:15, 2020.

－ Abstract

Outlines (closed loops) delineate areas of interest on terrains, such as regions with a heightened risk of landslides. For various analysis tasks it is necessary to define and compute a volume of earth (soil) based on such an outline, capturing, for example, the possible volume of a landslide in a high-risk region. In this paper we discuss several options to define meaningful 2D surfaces induced by a 1D outline, which allow us to compute such volumes. We experimentally compare the proposed surface options for two applications: Similarity of paths on terrains and landslide susceptibility analysis.
－ BibTeX
```
@article{volume-from-outlines-on-terrains:2020,
 title = {Volume from Outlines on Terrains},
 author = {Marc Van Kreveld and Tim Ophelders and Willem Sonke and Bettina Speckmann and Kevin Verbeek},
 year = {2020},
 bookTitle = {11th International Conference on Geographic Information Science, GIScience 2021},
}
```

Orthogonal Schematization with Minimum Homotopy Area
Bram A. Custers, Kevin A.B. Verbeek, Bettina Speckmann, Wouter Meulemans, Irina Kostitsyna, and Jeff Erickson.
2020.

－ BibTeX

@article{orthogonal-schematization-with-minimum-homotopy-area:2020,
    title = {Orthogonal Schematization with Minimum Homotopy Area},
    author = {Bram A. Custers and Kevin A.B. Verbeek and Bettina Speckmann and Wouter Meulemans and Irina Kostitsyna and Jeff Erickson},
    year = {2020},
    bookTitle = {},
}

2019

Computing Representative Networks for Braided Rivers
Maarten G. Kleinhans, Marc J. van Kreveld, Tim A.E. Ophelders, Willem M. Sonke, Bettina Speckmann, and Kevin A.B. Verbeek.
Journal of Computational Geometry, 10(1):423—443, 2019.

－ Abstract

Drainage networks on terrains have been studied extensively from an algorithmic perspective. However, in drainage networks water flow cannot bifurcate and hence they do not model braided rivers (multiple channels which split and join, separated by sediment bars). We initiate the algorithmic study of braided rivers by employing the descending quasi Morse-Smale complex on the river bed (a polyhedral terrain), and extending it with a certain ordering of bars from one river bank to the other. This allows us to compute a graph that models a representative channel network, consisting of lowest paths. To ensure that channels in this network are sufficiently different we define a sand function that represents the volume of sediment separating them. We show that in general the problem of computing a maximum network of non-crossing channels which are δ-different from each other (as measured by the sand function) is NP-hard. However, using our ordering between the river banks, we can compute a maximum δ-different network that respects this order in polynomial time. We implemented our approach and applied it to simulated and real-world braided rivers.
－ BibTeX
```
@article{computing-representative-networks-for-braided-rivers:2019,
 title = {Computing Representative Networks for Braided Rivers},
 author = {Maarten G. Kleinhans and Marc J. van Kreveld and Tim A.E. Ophelders and Willem M. Sonke and Bettina Speckmann and Kevin A.B. Verbeek},
 year = {2019},
 bookTitle = {Journal of Computational Geometry},
}
```
[ PDF ]
Locally Correct Fréchet Matchings
Kevin Buchin, Maike Buchin, Wouter Meulemans, and Bettina Speckmann.
Computational Geometry, 76:1—18, 2019.

－ Abstract

The Fréchet distance is a metric to compare two curves, which is based on monotone matchings between these curves. We call a matching that results in the Fréchet distance a Fréchet matching. There are often many different Fréchet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Fréchet matchings to “natural” matchings and to this end introduce locally correct Fréchet matchings. We prove that at least one such matching exists for two polygonal curves and give an O(N3log⁡N) algorithm to compute it, where N is the number of edges in both curves. We also present an O(N2) algorithm to compute a locally correct discrete Fréchet matching.
－ BibTeX
```
@article{locally-correct-fr-chet-matchings:2019,
 title = {Locally Correct Fréchet Matchings},
 author = {Kevin Buchin and Maike Buchin and Wouter Meulemans and Bettina Speckmann},
 year = {2019},
 bookTitle = {Computational Geometry},
}
```
[ PDF ]
Non-Crossing Paths with Geographic Constraints
Rodrigo I. Silveira, Bettina Speckmann, and Kevin Verbeek.
Discrete Mathematics and Theoretical Computer Science, 21(3):, 2019.

－ Abstract

A geographic network is a graph whose vertices are restricted to lie in a prescribed region in the plane. In this paper we begin to study the following fundamental problem for geographic networks: can a given geographic network be drawn without crossings? We focus on the seemingly simple setting where each region is a vertical segment, and one wants to connect pairs of segments with a path that lies inside the convex hull of the two segments. We prove that when paths must be drawn as straight line segments, it is NP-complete to determine if a crossing-free solution exists, even if all vertical segments have unit length. In contrast, we show that when paths must be monotone curves, the question can be answered in polynomial time. In the more general case of paths that can have any shape, we show that the problem is polynomial under certain assumptions.
－ BibTeX
```
@article{non-crossing-paths-with-geographic-constraints:2019,
 title = {Non-Crossing Paths with Geographic Constraints},
 author = {Rodrigo I. Silveira and Bettina Speckmann and Kevin Verbeek},
 year = {2019},
 bookTitle = {Discrete Mathematics and Theoretical Computer Science},
}
```
[ PDF ]
Optimal Morphs of Planar Orthogonal Drawings II
Arthur I. van Goethem, Kevin A.B. Verbeek, and Bettina Speckmann.
arXiv, 2019.

－ Abstract

Van Goethem and Verbeek recently described an algorithm that morphs between two planar orthogonal drawings $\Gamma_I$ and $\Gamma_O$ of a connected graph $G$ of complexity $n$, while preserving planarity, orthogonality, and linear complexity of the drawing during the morph. Necessarily drawings $\Gamma_I$ and $\Gamma_O$ must be equivalent; there exists a homeomorphism of the plane that transforms $\Gamma_I$ into $\Gamma_O$. The algorithm of van Goethem and Verbeek uses a linear number of linear morphs, however, if the graph $G$ is disconnected, then their method requires $O(n^{1.5})$ linear morphs. In this paper we present a refined version of their approach that allows us to also morph between two planar orthogonal drawings of a disconnected graph with a linear number of linear morphs while preserving planarity, orthogonality, and linear complexity. Van Goethem and Verbeek measure the structural difference between the two drawings in terms of the so-called \emph{spirality} $s = O(n)$ of $\Gamma_I$ relative to $\Gamma_O$ and describe a morph from $\Gamma_I$ to $\Gamma_O$ using $O(s)$ linear morphs. We show how to combine all intermittent morphs that act on links of spirality $s$ into one single linear morph and prove that $s+1$ linear morphs are always sufficient to morph between two planar orthogonal drawings, even for a disconnected graph. The resulting morphs are quite natural and visually pleasing.
－ BibTeX
```
@article{optimal-morphs-of-planar-orthogonal-drawings-ii:2019,
 title = {Optimal Morphs of Planar Orthogonal Drawings II},
 author = {Arthur I. van Goethem and Kevin A.B. Verbeek and Bettina Speckmann},
 year = {2019},
 bookTitle = {arXiv},
}
```
[ PDF ]
SolarView
Thom Castermans, Kevin Verbeek, Bettina Speckmann, Michel A. Westenberg, Rob Koopman, Shenghui Wang, Hein van den Berg, and Arianna Betti.
IEEE Transactions on Visualization and Computer Graphics, 25(10):2969—2982, 2019.

－ Abstract

We propose a novel type of low distortion radial embedding which focuses on one specific entity and its closest neighbors. Our embedding preserves near-exact distances to the focus entity and aims to minimize distortion between the other entities. We present an interactive exploration tool SolarView which places the focus entity at the center of a "solar system" and embeds its neighbors guided by concentric circles. SolarView provides an implementation of our novel embedding and several state-of-the-art dimensionality reduction and embedding techniques, which we adapted to our setting in various ways. We experimentally evaluated our embedding and compared it to these state-of-the-art techniques. The results show that our embedding competes with these techniques and achieves low distortion in practice. Our method performs particularly well when the visualization, and hence the embedding, adheres to the solar system design principle of our application. Nonetheless - as with all dimensionality reduction techniques - the distortion may be high. We leverage interaction techniques to give clear visual cues that allow users to accurately judge distortion. We illustrate the use of SolarView by exploring the high-dimensional metric space of bibliographic entity similarities.
－ BibTeX
```
@article{solarview:2019,
 title = {SolarView},
 author = {Thom Castermans and Kevin Verbeek and Bettina Speckmann and Michel A. Westenberg and Rob Koopman and Shenghui Wang and Hein van den Berg and Arianna Betti},
 year = {2019},
 bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
}
```
[ PDF ]
Spatially and Temporally Coherent Visual Summaries
Jules J.H.M. Wulms, Juri Buchmüller, Wouter Meulemans, Kevin A.B. Verbeek, and Bettina Speckmann.
arXiv, 2019:, 2019.

－ Abstract

When exploring large time-varying data sets, visual summaries are a useful tool to identify time intervals of interest for further consideration. A typical approach is to represent the data elements at each time step in a compact one-dimensional form or via a one-dimensional ordering. Such 1D representations can then be placed in temporal order along a time line. There are two main criteria to assess the quality of the resulting visual summary: spatial quality -- how well does the 1D representation capture the structure of the data at each time step, and stability -- how coherent are the 1D representations over consecutive time steps or temporal ranges? We focus on techniques that create such visual summaries using 1D orderings for entities moving in 2D. We introduce stable techniques based on well-established dimensionality-reduction techniques: Principle Component Analysis, Sammon mapping, and t-SNE. Our Stable Principal Component method is explicitly parametrized for stability, allowing a trade-off between the two quality criteria. We conduct computational experiments that compare our stable methods to various state-of-the-art approaches using a set of well-established quality metrics that capture the two main criteria. These experiments demonstrate that our stable algorithms outperform existing methods on stability, without sacrificing spatial quality or efficiency.
－ BibTeX
```
@article{spatially-and-temporally-coherent-visual-summaries:2019,
    title = {Spatially and Temporally Coherent Visual Summaries},
    author = {Jules J.H.M. Wulms and Juri Buchmüller and Wouter Meulemans and Kevin A.B. Verbeek and Bettina Speckmann},
    year = {2019},
    bookTitle = {arXiv},
}
```
[ PDF ]
A Practical Algorithm for Spatial Agglomerative Clustering
Thom Castermans, Bettina Speckmann, and Kevin Verbeek.
Proceedings of the Twenty-First Workshop on Algorithm Engineering and Experiments, pp. 174—185, 2019.

－ Abstract

 We study an agglomerative clustering problem motivated by visualizing disjoint glyphs (represented by geometric shapes) centered at specific locations on a geographic map. As we zoom out, the glyphs grow and start to overlap. We replace overlapping glyphs by one larger merged glyph to maintain disjointness. Our goal is to compute the resulting hierarchical clustering efficiently in practice. A straightforward algorithm for such spatial agglomerative clustering runs in On 2 log n time, where n is the number of glyphs. This is not efficient enough for many real-world datasets which contain up to tens or hundreds of thousands of glyphs. Recently the theoretical upper bound was improved to Onα(n) log 7 n time [10], where α(n) is the extremely slow growing inverse Ackermann function. Although this new algorithm is asymptotically much faster than the naive algorithm, from a practical point of view, it does not perform better for n ≤ 10 6 . In this paper we present a new agglomerative clustering algorithm which works efficiently in practice. Our algorithm relies on the use of quadtrees to speed up spatial computations. Interestingly, even in non-pathological datasets we can encounter large glyphs that intersect many quadtree cells and that are involved in many clustering events. We therefore devise a special strategy to handle such large glyphs. We test our algorithm on several synthetic and real-world datasets and show that it performs well in practice. 
－ BibTeX
```
@article{a-practical-algorithm-for-spatial-agglomerative-clustering:2019,
 title = {A Practical Algorithm for Spatial Agglomerative Clustering},
 author = {Thom Castermans and Bettina Speckmann and Kevin Verbeek},
 year = {2019},
 bookTitle = {Proceedings of the Twenty-First Workshop on Algorithm Engineering and Experiments},
}
```
[ PDF ]
Maximum Physically Consistent Trajectories
Bram Custers, Mees van de Kerkhof, Wouter Meulemans, Bettina Speckmann, and Frank Staals.
SIGSPATIAL '19: Proceedings of the 27th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pp. 79—88, 2019.

－ Abstract

Trajectories are usually collected with physical sensors, which are prone to errors and cause outliers in the data. We aim to identify such outliers via the physical properties of the tracked entity, that is, we consider its physical possibility to visit combinations of measurements. We describe optimal algorithms to compute maximum subsequences of measurements that are consistent with (simplified) physics models. Our results are output-sensitive with respect to the number k of outliers in a trajectory of n measurements. Specifically, we describe an O(n logn log 2k) time algorithm for 2D trajectories using a model with unbounded acceleration but bounded velocity, and an O(nk) time algorithm for any model where consistency is "concatenable": a consistent subsequence that ends where another begins together form a consistent sequence. We also consider acceleration-bounded models which are not concatenable. We show how to compute the maximum subsequence for such models in O(nk 2logk) time, under appropriate realism conditions. Finally, we experimentally explore the performance of our algorithms on several large real-world sets of trajectories. Our experiments show that we are generally able to retain larger fractions of noisy trajectories than previous work and simpler greedy approaches. We also observe that the speed-bounded model may in practice approximate the acceleration-bounded model quite well, though we observed some variation between datasets. 
－ BibTeX
```
@article{maximum-physically-consistent-trajectories:2019,
 title = {Maximum Physically Consistent Trajectories},
 author = {Bram Custers and Mees van de Kerkhof and Wouter Meulemans and Bettina Speckmann and Frank Staals},
 year = {2019},
 bookTitle = {SIGSPATIAL '19: Proceedings of the 27th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems},
}
```
Optimal Morphs of Planar Orthogonal Drawings II
Arthur van Goethem, Bettina Speckmann, and Kevin Verbeek.
Graph Drawing and Network Visualization - 27th International Symposium, GD 2019, Proceedings, pp. 33—45, 2019.

－ Abstract

Van Goethem and Verbeek [12] recently showed how to morph between two planar orthogonal drawings (Forumala Presented). of a connected graph G while preserving planarity, orthogonality, and the complexity of the drawing during the morph. Necessarily drawings (Forumala Presented). must be equivalent, that is, there exists a homeomorphism of the plane that transforms (Forumala Presented). Van Goethem and Verbeek use O(n) linear morphs, where n is the maximum complexity of the input drawings. However, if the graph is disconnected their method requires (Forumala Presented). linear morphs. In this paper we present a refined version of their approach that allows us to also morph between two planar orthogonal drawings of a disconnected graph with O(n) linear morphs while preserving planarity, orthogonality, and linear complexity of the intermediate drawings. Van Goethem and Verbeek measure the structural difference between the two drawings in terms of the so-called spirality (Forumala Presented). and describe a morph from (Forumala Presented). using O(s) linear morphs. We prove that linear morphs are always sufficient to morph between two planar orthogonal drawings, even for disconnected graphs. The resulting morphs are quite natural and visually pleasing.
－ BibTeX
```
@article{optimal-morphs-of-planar-orthogonal-drawings-ii:2019,
 title = {Optimal Morphs of Planar Orthogonal Drawings II},
 author = {Arthur van Goethem and Bettina Speckmann and Kevin Verbeek},
 year = {2019},
 bookTitle = {Graph Drawing and Network Visualization - 27th International Symposium, GD 2019, Proceedings},
}
```
Preprocessing Ambiguous Imprecise Points
Ivor van der Hoog, Irina Kostitsyna, Maarten Löffler, and Bettina Speckmann.
35th International Symposium on Computational Geometry (SoCG 2019), 129:42:1—42:16, 2019.

－ Abstract

Let R = {R 1,R 2, . . . ,R n} be a set of regions and let X = {x 1, x 2, . . . , x n} be an (unknown) point set with x i ϵ R i. Region Ri represents the uncertainty region of x i. We consider the following question: how fast can we establish order if we are allowed to preprocess the regions in R? The preprocessing model of uncertainty uses two consecutive phases: a preprocessing phase which has access only to R followed by a reconstruction phase during which a desired structure on X is computed. Recent results in this model parametrize the reconstruction time by the ply of R, which is the maximum overlap between the regions in R. We introduce the ambiguity A(R) as a more fine-grained measure of the degree of overlap in R. We show how to preprocess a set of d-dimensional disks in O(n log n) time such that we can sort X (if d = 1) and reconstruct a quadtree on X (if d ≥ 1 but constant) in O(A(R)) time. If A(R) is sub-linear, then reporting the result dominates the running time of the reconstruction phase. However, we can still return a suitable data structure representing the result in O(A(R)) time. In one dimension, R is a set of intervals and the ambiguity is linked to interval entropy, which in turn relates to the well-studied problem of sorting under partial information. The number of comparisons necessary to find the linear order underlying a poset P is lower-bounded by the graph entropy of P. We show that if P is an interval order, then the ambiguity provides a constant-factor approximation of the graph entropy. This gives a lower bound of (A(R)) in all dimensions for the reconstruction phase (sorting or any proximity structure), independent of any preprocessing; hence our result is tight. Finally, our results imply that one can approximate the entropy of interval graphs in O(n log n) time, improving the O(n 2.5) bound by Cardinal et al. 
－ BibTeX
```
@article{preprocessing-ambiguous-imprecise-points:2019,
 title = {Preprocessing Ambiguous Imprecise Points},
 author = {Ivor van der Hoog and Irina Kostitsyna and Maarten Löffler and Bettina Speckmann},
 year = {2019},
 bookTitle = {35th International Symposium on Computational Geometry (SoCG 2019)},
}
```
[ PDF ]
SETH Says: Weak Fréchet Distance is Faster, but Only if It is Continuous and in One Dimension
Kevin Buchin, Tim Ophelders, and Bettina Speckmann.
Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2887—2899, 2019.

－ Abstract

We show by reduction from the Orthogonal Vectors problem that algorithms with strongly subquadratic running time cannot approximate the Fréchet distance between curves better than a factor 3 unless SETH fails. We show that similar reductions cannot achieve a lower bound with a factor better than 3. Our lower bound holds for the continuous, the discrete, and the weak discrete Fréchet distance even for curves in one dimension. Interestingly, the continuous weak Fréchet distance behaves differently. Our lower bound still holds for curves in two dimensions and higher. However, for curves in one dimension, we provide an exact algorithm to compute the weak Fréchet distance in linear time.
－ BibTeX
```
@article{seth-says-weak-fr-chet-distance-is-faster-but-only-if-it-is-continuous-and-in-one-dimension:2019,
 title = {SETH Says: Weak Fréchet Distance is Faster, but Only if It is Continuous and in One Dimension},
 author = {Kevin Buchin and Tim Ophelders and Bettina Speckmann},
 year = {2019},
 bookTitle = {Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)},
}
```
Kinetic Volume-Based Persistence for 1D Terrains
Tim Ophelders, Willem Sonke, Bettina Speckmann, and Kevin Verbeek.
pp. 38:1—38:7, 2019.

－ Abstract

Persistence is the method of choice to simplify terrains by removing insignificant features while retaining topologically important ones. Motivated by applications in geomorphology, we study volume-persistence, a variant of persistence which is based on the volume underneath the terrain (instead of the usual vertex heights). Specifically, we want to kinetically maintain a volume-simplified time-varying terrain. In this paper we describe a kinetic data structure (KDS) that maintains the pruned split tree of an area-persistent 1D terrain under linear vertex motion. The main ingredient of this KDS is an algorithm that detects those combinatorial events when a pruned part of the terrain attains a certain threshold volume.
－ BibTeX
```
@article{kinetic-volume-based-persistence-for-1d-terrains:2019,
    title = {Kinetic Volume-Based Persistence for 1D Terrains},
    author = {Tim Ophelders and Willem Sonke and Bettina Speckmann and Kevin Verbeek},
    year = {2019},
    bookTitle = {},
}
```
[ PDF ]
Maximum Physically Consistent Trajectories
Bram Custers, Mees van de Kerkhof, Wouter Meulemans, Bettina Speckmann, and Frank Staals.
2019.

－ Abstract

We study the problem of detecting outlying measurements in a GPS trajectory. Our method considers the physical possibility for the tracked object to visit combinations of measurements,using simplified physics models. We aim to compute the maximum subsequence of the measurements that is consistent with a given physics model. We give an O(n log³ n) time algorithm for 2D-trajectories in a model with unbounded acceleration but bounded velocity, and an output-sensitive algorithm for any model where consistency checks can be done in O(1) time and consistency is transitive.
－ BibTeX
```
@article{maximum-physically-consistent-trajectories:2019,
    title = {Maximum Physically Consistent Trajectories},
    author = {Bram Custers and Mees van de Kerkhof and Wouter Meulemans and Bettina Speckmann and Frank Staals},
    year = {2019},
    bookTitle = {},
}
```
[ PDF ]

2018

Colored Spanning Graphs for Set Visualization
F. Hurtado, M. Korman, M.J. van Kreveld, M. Löffler, V. Sacristán, A. Shioura, R.I. Silveira, B. Speckmann, and T. Tokuyama.
Computational Geometry, 68:262—276, 2018.

－ Abstract

We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A red-blue-purple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem can be solved in polynomial time using matroid techniques. In addition, we discuss more efficient algorithms for the case in which points are located on a line or a circle, and also describe a fast ([Formula presented]ρ+1)-approximation algorithm, where ρ is the Steiner ratio.
－ BibTeX
```
@article{colored-spanning-graphs-for-set-visualization:2018,
 title = {Colored Spanning Graphs for Set Visualization},
 author = {F. Hurtado and M. Korman and M.J. van Kreveld and M. Löffler and V. Sacristán and A. Shioura and R.I. Silveira and B. Speckmann and T. Tokuyama},
 year = {2018},
 bookTitle = {Computational Geometry},
}
```
[ PDF ]
Computing the Similarity Between Moving Curves
K. Buchin, T. Ophelders, and B. Speckmann.
Computational Geometry, 73:2—14, 2018.

－ Abstract

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fréchet distance between surfaces. While the Fréchet distance between surfaces is generally NP-hard, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on a relation between obstacles.
－ BibTeX
```
@article{computing-the-similarity-between-moving-curves:2018,
 title = {Computing the Similarity Between Moving Curves},
 author = {K. Buchin and T. Ophelders and B. Speckmann},
 year = {2018},
 bookTitle = {Computational Geometry},
}
```
[ PDF ]
Homotopic c-Oriented Routing with Few Links and Thick Edges
B. Speckmann and K.A.B. Verbeek.
Computational Geometry, 67:11—28, 2018.

－ Abstract

We study the NP-hard optimization problem of finding non-crossing thick C-oriented paths that are homotopic to a set of input paths in an environment with C-oriented obstacles, with the goal to minimize the total number of links of the paths. We introduce a special type of C-oriented paths—smooth paths—and present a 2-approximation algorithm for smooth paths that runs in O(n3log⁡κ+kinlog⁡n+kout) time, where n is the total number of paths and obstacle vertices, kin and kout are the total complexities of the input and output paths, and κ=|C|. The algorithm also computes an O(κ)-approximation for general C-oriented paths. In particular we give a 2-approximation algorithm for rectilinear paths. Our algorithm not only approximates the minimum number of links, but also simultaneously minimizes the total length of the paths. As a related result we show that, given a set of (possibly crossing) C-oriented paths with a total of L links, non-crossing C-oriented paths homotopic to the input paths can require a total of Ω(Llog⁡κ) links.
－ BibTeX
```
@article{homotopic-c-oriented-routing-with-few-links-and-thick-edges:2018,
 title = {Homotopic c-Oriented Routing with Few Links and Thick Edges},
 author = {B. Speckmann and K.A.B. Verbeek},
 year = {2018},
 bookTitle = {Computational Geometry},
}
```
Simultaneous Visualization of Language Endangerment and Language Description
Harald Hammarström, T.H.A. Castermans, Robert Forkel, Kevin Verbeek, Michel A. Westenberg, and Bettina Speckmann.
Language Documentation & Conservation, 12:359—392, 2018.

－ Abstract

The world harbors a diversity of some 6,500 mutually unintelligible languages. As has been increasingly observed by linguists, many minority languages are becoming endangered and will be lost forever if not documented. Urgently indeed, many efforts are being launched to document and describe languages. This undertaking naturally has the priority toward the most endangered and least described languages. For the first time, we combine world-wide databases on language description (Glottolog) and language endangerment (ElCat, Ethnologue, UNESCO) and provide two online interfaces, GlottoScope and GlottoVis, to visualize these together. The interfaces are capable of browsing, filtering, zooming, basic statistics, and different ways of combining the two measures on a world map background. GlottoVis provides advanced techniques for combining cluttered dots on a map. With the tools and databases described we seek to increase the overall knowledge of the actual state language endangerment and description worldwide.
－ BibTeX
```
@article{simultaneous-visualization-of-language-endangerment-and-language-description:2018,
    title = {Simultaneous Visualization of Language Endangerment and Language Description},
    author = {Harald Hammarström and T.H.A. Castermans and Robert Forkel and Kevin Verbeek and Michel A. Westenberg and Bettina Speckmann},
    year = {2018},
    bookTitle = {Language Documentation & Conservation},
}
```
[ PDF ]
Stable Treemaps Via Local Moves
M. Sondag, B. Speckmann, and K.A.B. Verbeek.
IEEE Transactions on Visualization and Computer Graphics, 24(1):729—738, 2018.

－ Abstract

Treemaps are a popular tool to visualize hierarchical data: items are represented by nested rectangles and the area of each rectangle corresponds to the data being visualized for this item. The visual quality of a treemap is commonly measured via the aspect ratio of the rectangles. If the data changes, then a second important quality criterion is the stability of the treemap: how much does the treemap change as the data changes. We present a novel stable treemapping algorithm that has very high visual quality. Whereas existing treemapping algorithms generally recompute the treemap every time the input changes, our algorithm changes the layout of the treemap using only local modifications. This approach not only gives us direct control over stability, but it also allows us to use a larger set of possible layouts, thus provably resulting in treemaps of higher visual quality compared to existing algorithms. We further prove that we can reach all possible treemap layouts using only our local modifications. Furthermore, we introduce a new measure for stability that better captures the relative positions of rectangles. We finally show via experiments on real-world data that our algorithm outperforms existing treemapping algorithms also in practice on either visual quality and/or stability. Our algorithm scores high on stability regardless of whether we use an existing stability measure or our new measure.
－ BibTeX
```
@article{stable-treemaps-via-local-moves:2018,
 title = {Stable Treemaps Via Local Moves},
 author = {M. Sondag and B. Speckmann and K.A.B. Verbeek},
 year = {2018},
 bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
}
```
[ PDF ]
A Framework for Algorithm Stability and Its Application to Kinetic Euclidean MSTs
Wouter Meulemans, Bettina Speckmann, Kevin Verbeek, and Jules Wulms.
LATIN 2018: Theoretical Informatics, pp. 805—819, 2018.

－ Abstract

We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms. In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees.
－ BibTeX
```
@article{a-framework-for-algorithm-stability-and-its-application-to-kinetic-euclidean-msts:2018,
 title = {A Framework for Algorithm Stability and Its Application to Kinetic Euclidean MSTs},
 author = {Wouter Meulemans and Bettina Speckmann and Kevin Verbeek and Jules Wulms},
 year = {2018},
 bookTitle = {LATIN 2018: Theoretical Informatics},
}
```
[ PDF ]
A Philosophical Perspective on Visualization for Digital Humanities
Hein van den Berg, Arianna Betti, Thom Castermans, Rob Koopman, Bettina Speckmann, Kevin Verbeek, Titia van der Werf, Shenghui Wang, and Michel A. Westenberg.
Proc. 3rd Workshop on Visualization for the Digital Humanities (VIS4DH), 2018.

－ Abstract

In this position paper, we describe a number of methodological and philosophical challenges that arose within our interdisciplinary Digital Humanities project \catvis, which is a collaboration between applied geometric algorithms and visualization researchers, data scientists working at OCLC, and philosophers who have a strong interest in the methodological foundations of visualization research. The challenges we describe concern aspects of one single epistemic need: that of methodologically securing (an increase in) trust in visualizations. We discuss the lack of ground truths in the (digital) humanities and argue that trust in visualizations requires that we evaluate visualizations on the basis of ground truths that humanities scholars themselves create. We further argue that trust in visualizations requires that a visualization provides provable guarantees on the faithfulness of the visual representation and that we must clearly communicate to the users which part of the visualization can be trusted and how much. Finally, we discuss transparency and accessibility in visualization research and provide measures for securing transparency and accessibility.
－ BibTeX
```
@article{a-philosophical-perspective-on-visualization-for-digital-humanities:2018,
    title = {A Philosophical Perspective on Visualization for Digital Humanities},
    author = {Hein van den Berg and Arianna Betti and Thom Castermans and Rob Koopman and Bettina Speckmann and Kevin Verbeek and Titia van der Werf and Shenghui Wang and Michel A. Westenberg},
    year = {2018},
    bookTitle = {Proc. 3rd Workshop on Visualization for the Digital Humanities (VIS4DH)},
}
```
[ PDF ]
Agglomerative Clustering of Growing Squares
Thom Castermans, Bettina Speckmann, Frank Staals, and Kevin Verbeek.
13th Latin American Theoretical INformatics Symposium (LATIN), pp. 260—274, 2018.

－ Abstract

We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of n disjoint growing squares. Our data structure uses O(n(log n log log n)2) space, supports queries in worst case O(log3 n) time, and updates in O(log7 n) amortized time. This leads to an O(n α(n) log7 n) time algorithm (where α is the inverse Ackermann function) to solve the agglomerative clustering problem, which is a significant improvement over the straightforward O(n2 log n) time algorithm.
－ BibTeX
```
@article{agglomerative-clustering-of-growing-squares:2018,
 title = {Agglomerative Clustering of Growing Squares},
 author = {Thom Castermans and Bettina Speckmann and Frank Staals and Kevin Verbeek},
 year = {2018},
 bookTitle = {13th Latin American Theoretical INformatics Symposium (LATIN)},
}
```
[ PDF ]

Foreword
Bettina Speckmann, Csaba D. Tóth, and Xavier Goaoc.
34th International Symposium on Computational Geometry, SoCG 2018; Budapest; Hungary; 11 June 2018 through 14 June 2018, 99:xi, 2018.

－ BibTeX

@article{foreword:2018,
    title = {Foreword},
    author = {Bettina Speckmann and Csaba D. Tóth and Xavier Goaoc},
    year = {2018},
    bookTitle = {34th International Symposium on Computational Geometry, SoCG 2018; Budapest; Hungary; 11 June 2018 through 14 June 2018},
}

[ PDF ]

Non-Crossing Paths with Geographic Constraints
R.I. Silveira, B. Speckmann, and K.A.B. Verbeek.
Graph drawing and network visualization - 25th International Symposium, GD 2017, Revised Selected Papers, pp. 454—461, 2018.

－ Abstract

A geographic network is a graph whose vertices are restricted to lie in a prescribed region in the plane. In this paper we begin to study the following fundamental problem for geographic networks: can a given geographic network be drawn without crossings? We focus on the seemingly simple setting where each region is a unit length vertical segment, and one wants to connect pairs of segments with a path that lies inside the convex hull of the two segments. We prove that when paths must be drawn as straight line segments, it is NP-complete to determine if a crossing-free solution exists. In contrast, we show that when paths must be monotone curves, the question can be answered in polynomial time. In the more general case of paths that can have any shape, we show that the problem is polynomial under certain assumptions.
－ BibTeX
```
@article{non-crossing-paths-with-geographic-constraints:2018,
 title = {Non-Crossing Paths with Geographic Constraints},
 author = {R.I. Silveira and B. Speckmann and K.A.B. Verbeek},
 year = {2018},
 bookTitle = {Graph drawing and network visualization - 25th International Symposium, GD 2017, Revised Selected Papers},
}
```
Optimal Algorithms for Compact Linear Layouts
Willem Sonke, Kevin Verbeek, Wouter Meulemans, Eric Verbeek, and Bettina Speckmann.
2018 IEEE Pacific Visualization Symposium, PacificVis 2018, pp. 1—10, 2018.

－ Abstract

Linear layouts are a simple and natural way to draw a graph: all vertices are placed on a single line and edges are drawn as arcs between the vertices. Despite its simplicity, a linear layout can be a very meaningful visualization if there is a particular order defined on the vertices. Common examples of such ordered - and often also directed - graphs are event sequences and processes. A main drawback of linear layouts are the usually (very) large aspect ratios of the resulting drawings, which prevent users from obtaining a good overview of the whole graph. In this paper we present a novel and versatile algorithm to optimally fold a linear layout of a graph such that it can be drawn nicely in a specified aspect ratio, while still clearly communicating the linearity of the layout. Our algorithm allows vertices to be drawn as blocks or rectangles of specified sizes to incorporate different drawing styles, label sizes, and even recursive structures. For reasonably-sized drawings the folded layout can be computed interactively. We demonstrate the applicability of our algorithm on graphs that represent process trees, a particular type of process model. Our algorithm arguably produces much more readable layouts than existing methods.
－ BibTeX
```
@article{optimal-algorithms-for-compact-linear-layouts:2018,
 title = {Optimal Algorithms for Compact Linear Layouts},
 author = {Willem Sonke and Kevin Verbeek and Wouter Meulemans and Eric Verbeek and Bettina Speckmann},
 year = {2018},
 bookTitle = {2018 IEEE Pacific Visualization Symposium, PacificVis 2018},
}
```

Social Network-Epistemology
M. Alfano, S. Cunningham, W. Meulemans, I. Rutter, M. Sondag, B. Speckmann, and E. Sullivan.
Proceedings - IEEE 14th International Conference on eScience, e-Science 2018, pp. 320—321, 2018.

－ BibTeX

@article{social-network-epistemology:2018,
    title = {Social Network-Epistemology},
    author = {M. Alfano and S. Cunningham and W. Meulemans and I. Rutter and M. Sondag and B. Speckmann and E. Sullivan},
    year = {2018},
    bookTitle = {Proceedings - IEEE 14th International Conference on eScience, e-Science 2018},
}

Volume-Based Similarity of Linear Features on Terrains
Willem Sonke, Marc van Kreveld, Tim Ophelders, Bettina Speckmann, and Kevin Verbeek.
26th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, ACM SIGSPATIAL GIS 2018, pp. 444—447, 2018.

－ Abstract

Linear features on terrains model the boundaries of ground cover regions, delineate glaciers, or form the boundary of rivers and lakes. When computing the similarity between such linear features, it is important to also take their context into account: the terrain. We hence explore the possibilities of volume-based distance measures for linear features on a terrain. Our measures construct suitable base surfaces between the linear features, which can slice through the input terrain and also hover above. The similarity between two linear features is then captured by the volume of “earth” above the base surface and below the terrain, and possibly also by the volume of “air” below the base surface and above the terrain. We suggest six ways of choosing a suitable base surface. These choices give rise to different measured volumes and can be useful in different application scenarios.
－ BibTeX
```
@article{volume-based-similarity-of-linear-features-on-terrains:2018,
 title = {Volume-Based Similarity of Linear Features on Terrains},
 author = {Willem Sonke and Marc van Kreveld and Tim Ophelders and Bettina Speckmann and Kevin Verbeek},
 year = {2018},
 bookTitle = {26th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, ACM SIGSPATIAL GIS 2018},
}
```
[ PDF ]
A Framework for Algorithm Stability and Its Application to Kinetic Euclidean MSTs
W. Meulemans, B. Speckmann, K.A.B. Verbeek, and J.J.H.M. Wulms.
pp. 11:1—11:6, 2018.

－ Abstract

We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms. In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. We demonstrate the use of topological stability by applying it to kinetic Euclidean minimum spanning trees.
－ BibTeX
```
@article{a-framework-for-algorithm-stability-and-its-application-to-kinetic-euclidean-msts:2018,
 title = {A Framework for Algorithm Stability and Its Application to Kinetic Euclidean MSTs},
 author = {W. Meulemans and B. Speckmann and K.A.B. Verbeek and J.J.H.M. Wulms},
 year = {2018},
 bookTitle = {},
}
```
[ PDF ]
A KDS for Discrete Morse-Smale Complexes
Tim Ophelders, Willem Sonke, Bettina Speckmann, and Kevin Verbeek.
pp. 3:1—3:2, 2018.

－ Abstract

The Morse-Smale complex of a terrain is a topological complex that provides information about the features of the terrain. It consists of the critical points (minima, saddles and maxima), together with steepest-descent paths from saddles to minima and steepest-ascent paths from saddles to maxima. We describe a kinetic data structure to maintain the Morse-Smale-complex for a triangulated terrain whose vertex heights change continuously. This can be used to efficiently analyze time-varying data.
－ BibTeX
```
@article{a-kds-for-discrete-morse-smale-complexes:2018,
    title = {A KDS for Discrete Morse-Smale Complexes},
    author = {Tim Ophelders and Willem Sonke and Bettina Speckmann and Kevin Verbeek},
    year = {2018},
    bookTitle = {},
}
```
[ PDF ]

Agglomerative Clustering of Growing Squares
Thom Castermans, Bettina Speckmann, Frank Staals, and Kevin Verbeek.
pp. 8:1—8:6, 2018.

－ BibTeX

@article{agglomerative-clustering-of-growing-squares:2018,
    title = {Agglomerative Clustering of Growing Squares},
    author = {Thom Castermans and Bettina Speckmann and Frank Staals and Kevin Verbeek},
    year = {2018},
    bookTitle = {},
}

[ PDF ]

Geometry and Topology of Estuary and Braided River Channel Networks Extracted from Topographic Data
Matthew Hiatt, Willem Sonke, Elisabeth Addink, Wout van Dijk, Marc J. van Kreveld, Tim Ophelders, Kevin Verbeek, Joyce Vlaming, Bettina Speckmann, and Maarten G. Kleinhans.
2018.

－ Abstract

Channels are ubiquitous features of Earth's surface that are important pathways for the transport of water, solids, and solutes across landscapes, provide a range of ecosystem services, and support economic activity. Networks are mathematical representations of the connections among a set of objects and are useful representations of topology, geometry, and connectivity in channelized environments. However, objective and automatic extraction of channel networks from topography in multi-channel systems like braided river and estuaries has remained elusive. We present a mathematically-rigorous framework from extracting network topology and geometry from digital elevation models (DEMs) of braided rivers and estuaries. The concept of the “sand function” is introduced, which quantifies the volume of material separating channels and is a useful metric for identifying the relative scales of channels in the network. Four case studies are included: DEMs from the Western Scheldt estuary (Netherlands) and the Waimakariri River (New Zealand), as well as DEMs generated by numerical models of the morphodynamics in a braided river and an estuary. We show that larger scale channels (with higher sand function values) in the estuaries tend to be significantly deeper than smaller scale channels in the network. The results suggest that the main channel in an estuary is significantly deeper than the rest of the network, while the braided rivers tend to have channel depths that are evenly-distributed across channel scales. In all cases, the length of channels relative to system size scales with sand function scale to the power of 0.24-0.35, while the number of nodes against system scale does not exhibit a power-law relationship. The methods and results presented in this study provide a benchmark for evaluating both geometric and topologic characteristics of multi-threaded channel networks across scales.
－ BibTeX
```
@article{geometry-and-topology-of-estuary-and-braided-river-channel-networks-extracted-from-topographic-data:2018,
    title = {Geometry and Topology of Estuary and Braided River Channel Networks Extracted from Topographic Data},
    author = {Matthew Hiatt and Willem Sonke and Elisabeth Addink and Wout van Dijk and Marc J. van Kreveld and Tim Ophelders and Kevin Verbeek and Joyce Vlaming and Bettina Speckmann and Maarten G. Kleinhans},
    year = {2018},
    bookTitle = {},
}
```
Optimal Algorithms for Compact Linear Layouts
Wouter Meulemans, Willem Sonke, Bettina Speckmann, Eric Verbeek, and Kevin Verbeek.
pp. 10:1—10:6, 2018.

－ Abstract

Linear layouts are a simple and natural way to draw a graph: all vertices are placed on a single line and edges are drawn as arcs between the vertices. Despite its simplicity, a linear layout can be a very meaningful visualization if there is a particular order defined on the vertices. Common examples of such ordered - and often also directed - graphs are event sequences and processes: public transport systems tracking passenger check-in and check-out, banks checking online transactions, or hospitals recording the paths of patients through their system, to name a few. A main drawback of linear layouts are the usually (very) large aspect ratios of the resulting drawings, which prevent users from obtaining a good overview of the whole graph. In this paper we present a novel and versatile algorithm to optimally fold a linear layout of a graph such that it can be drawn effectively in a specified aspect ratio, while still clearly communicating the linearity of the layout.
－ BibTeX
```
@article{optimal-algorithms-for-compact-linear-layouts:2018,
    title = {Optimal Algorithms for Compact Linear Layouts},
    author = {Wouter Meulemans and Willem Sonke and Bettina Speckmann and Eric Verbeek and Kevin Verbeek},
    year = {2018},
    bookTitle = {},
}
```
[ PDF ]
The Effects of Dredging and Disposal Activity on the Resilience of Estuary Morphodynamics
Wout van Dijk, Jasper Leuven, Jana Cox, Jelmer Cleveringa, Marcel Taal, Matthew Hiatt, Willem Sonke, Kevin Verbeek, Bettina Speckmann, and Maarten G. Kleinhans.
2018.

－ Abstract

Shipping fairways in estuaries are continuously dredged to maintain access to major ports for large ships. However, various estuaries worldwide show adverse side effects to dredging activities, including shifts from a multi-channel system to a single channel, loss of ecologically-valuable intertidal areas, and increased muddiness. The Western Scheldt (Netherlands) is an example of a system where several million m3 sediments are dredged annually and disposed of within the estuary. Here, the effects of dredging and disposal strategies on the channel-shoal morphodynamics, including the evolution of the multi-channel system and its ecological impact are studied using field observations, numerical and physical models. Time series of bathymetry, morphodynamic model runs and physical experiments in the Metronome tilting flume show for the first time that the deliberate dredging and disposal strategy increases the surface area of inter-tidal habitats, which decreases the connectivity between channels and destabilizes the multi-channel system. We quantify these changes in scale and topology using a novel and mathematically-rigorous network extraction, which identifies the partial switching of the main channel with the side channel in the past and that continuous disposal of sediment in the side channel results in siltation. The physical experiments indicate that the estuarine network is dramatically changed by dredging even for a long period following the halting of any dredging and dumping works. The morphodynamic model similarly shows that dredging of the sills is essential to maintain the current shipping fairway. A new disposal strategy targeting the deepest scours of the main channel appears to be the optimal solution for ecological value and sustains dynamic channel-shoal interactions and the multi-channel pattern. Sea level rise scenarios demonstrate the importance of keeping the sediment in the system for sustainable management, but present dredging and disposal techniques put current ecosystems under pressure.
－ BibTeX
```
@article{the-effects-of-dredging-and-disposal-activity-on-the-resilience-of-estuary-morphodynamics:2018,
    title = {The Effects of Dredging and Disposal Activity on the Resilience of Estuary Morphodynamics},
    author = {Wout van Dijk and Jasper  Leuven and Jana Cox and Jelmer Cleveringa and Marcel Taal and Matthew Hiatt and Willem Sonke and Kevin Verbeek and Bettina Speckmann and Maarten G. Kleinhans},
    year = {2018},
    bookTitle = {},
}
```

Preface
Magnús M. Halldórsson, Naoki Kobayashi, and Bettina Speckmann.
Information and Computation, 261(2):159, 2018.

－ BibTeX

@article{preface:2018,
    title = {Preface},
    author = {Magnús M. Halldórsson and Naoki Kobayashi and Bettina Speckmann},
    year = {2018},
    bookTitle = {Information and Computation},
}

[ PDF ]

2017

A Framework for Algorithm Stability
W. Meulemans, B. Speckmann, K.A.B. Verbeek, and J. Wulms.
arXiv, 2017.

－ Abstract

We say that an algorithm is stable if small changes in the input result in small changes in the output. Algorithm stability plays an important role when analyzing and visualizing time-varying data. However, so far, there are only few theoretical results on the stability of algorithms, possibly due to a lack of theoretical analysis tools. In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes. Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability. We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees.
－ BibTeX
```
@article{a-framework-for-algorithm-stability:2017,
    title = {A Framework for Algorithm Stability},
    author = {W. Meulemans and B. Speckmann and K.A.B. Verbeek and J. Wulms},
    year = {2017},
    bookTitle = {arXiv},
}
```
[ PDF ]
Column Planarity and Partially-Simultaneous Geometric Embedding
L. Barba, W.S. Evans, M.H.W. Hoffmann, V. Kusters, M. Saumell, and B. Speckmann.
Journal of Graph Algorithms and Applications, 21(6):983—1002, 2017.

－ Abstract

We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for the maximum size of column planar subsets of trees: every tree on n vertices contains a column planar set of size at least 14n/17 and for any ɛ > 0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at most (5/6 + ɛ)n. In addition, we show that every outerplanar graph has a column planar set of size at least n/2. We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partially-simultaneous geometric embedding (PSGE). A PSGE of two graphs G1 and G2 allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct k-PSGEs, which are PSGEs in which at least k vertices are mapped to the same point for both graphs. In particular, we show that every two trees on n vertices admit an 11n/17-PSGE and every two outerplanar graphs admit an n/4-PSGE.
－ BibTeX
```
@article{column-planarity-and-partially-simultaneous-geometric-embedding:2017,
 title = {Column Planarity and Partially-Simultaneous Geometric Embedding},
 author = {L. Barba and W.S. Evans and M.H.W. Hoffmann and V. Kusters and M. Saumell and B. Speckmann},
 year = {2017},
 bookTitle = {Journal of Graph Algorithms and Applications},
}
```
[ PDF ]
Multi-Granular Trend Detection for Time-Series Analysis
A.I. van Goethem, F. Staals, M. Löffler, J. Dykes, and B. Speckmann.
IEEE Transactions on Visualization and Computer Graphics, 23(1):661—670, 2017.

－ Abstract

Time series (such as stock prices) and ensembles (such as model runs for weather forecasts) are two important types of one-dimensional time-varying data. Such data is readily available in large quantities but visual analysis of the raw data quickly becomes infeasible, even for moderately sized data sets. Trend detection is an effective way to simplify time-varying data and to summarize salient information for visual display and interactive analysis. We propose a geometric model for trend-detection in one-dimensional time-varying data, inspired by topological grouping structures for moving objects in two- or higher-dimensional space. Our model gives provable guarantees on the trends detected and uses three natural parameters: granularity, support-size, and duration. These parameters can be changed on-demand. Our system also supports a variety of selection brushes and a time-sweep to facilitate refined searches and interactive visualization of (sub-)trends. We explore different visual styles and interactions through which trends, their persistence, and evolution can be explored.
－ BibTeX
```
@article{multi-granular-trend-detection-for-time-series-analysis:2017,
    title = {Multi-Granular Trend Detection for Time-Series Analysis},
    author = {A.I. van Goethem and F. Staals and M. Löffler and J. Dykes and B. Speckmann},
    year = {2017},
    bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
}
```
[ PDF ]
Packing Plane Spanning Trees and Paths in Complete Geometric Graphs
O. Aichholzer, T. Hackl, M. Korman, M.J. van Kreveld, M. Löffler, A. Pilz, B. Speckmann, and E. Welzl.
Information Processing Letters, 124:35—41, 2017.

－ Abstract

We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph GKn on any set S of n points in general position in the plane? We show that this number is in Ω(n). Further, we consider variants of this problem by bounding the diameter and the degree of the trees (in particular considering spanning paths).
－ BibTeX
```
@article{packing-plane-spanning-trees-and-paths-in-complete-geometric-graphs:2017,
 title = {Packing Plane Spanning Trees and Paths in Complete Geometric Graphs},
 author = {O. Aichholzer and T. Hackl and M. Korman and M.J. van Kreveld and M. Löffler and A. Pilz and B. Speckmann and E. Welzl},
 year = {2017},
 bookTitle = {Information Processing Letters},
}
```
[ PDF ]
Complexity Measures for Mosaic Drawings
Q.W. Bouts, B. Speckmann, and K.A.B. Verbeek.
WALCOM: Algorithms and Computation - 11th International Conference and Workshops, WALCOM 2017, Proceedings, pp. 149—160, 2017.

－ Abstract

Graph Drawing uses a well established set of complexity measures to determine the quality of a drawing, most notably the area of the drawing and the complexity of the edges. For contact representations the complexity of the shapes representing vertices also clearly contributes to the complexity of the drawing. Furthermore, if a contact representation does not fill its bounding shape completely, then also the complexity of its complement is visually salient. We study the complexity of contact representations with variable shapes, specifically mosaic drawings.Mosaic drawings are drawn on a tiling of the plane and represent vertices by configurations: simply-connected sets of tiles. The complement of a mosaic drawing with respect to its bounding rectangle is also a set of simply-connected tiles, the channels. We prove that simple mosaic drawings without channels may require Ω(n2) area. This bound is tight. If we use only straight channels, then outerplanar graphs with k ears may require Ω(min(nk, n2/k)) area. This bound is partially tight: we show how to draw outerplanar graphs with k ears in O(nk) area with L-shaped vertex configurations and straight channels. Finally, we argue that L-shaped channels are strictly more powerful than straight channels, but may still require Ω(n7/6) area.
－ BibTeX
```
@article{complexity-measures-for-mosaic-drawings:2017,
 title = {Complexity Measures for Mosaic Drawings},
 author = {Q.W. Bouts and B. Speckmann and K.A.B. Verbeek},
 year = {2017},
 bookTitle = {WALCOM: Algorithms and Computation - 11th International Conference and Workshops, WALCOM 2017, Proceedings},
}
```
[ PDF ]
Computing Optimal Homotopies over a Spiked Plane with Polygonal Boundary
B. Burton, E.W. Chambers, M.J. Van Kreveld, W. Meulemans, T.A.E. Ophelders, and B. Speckmann.
Proc. 25th European Symposium on Algorithms (ESA), pp. 1—14, 2017.

－ Abstract

Computing optimal deformations between two curves is a fundamental question with various applications, and has recently received much attention in both computational topology and in mathematics in the form of homotopies of disks and annular regions. In this paper, we examine this problem in a geometric setting, where we consider the boundary of a polygonal domain with spikes, point obstacles that can be crossed at an additive cost. We aim to continuously morph from one part of the boundary to another, necessarily passing over all spikes, such that the most expensive intermediate curve is minimized, where the cost of a curve is its geometric length plus the cost of any spikes it crosses. We first investigate the general setting where each spike may have a different cost. For the number of inflection points in an intermediate curve, we present a lower bound that is linear in the number of spikes, even if the domain is convex and the two boundaries for which we seek a morph share an endpoint. We describe a 2-Approximation algorithm for the general case, and an optimal algorithm for the case that the two boundaries for which we seek a morph share both endpoints, thereby representing the entire boundary of the domain. We then consider the setting where all spikes have the same unit cost and we describe a polynomial-Time exact algorithm. The algorithm combines structural properties of homotopies arising from the geometry with methodology for computing Fréchet distances.
－ BibTeX
```
@article{computing-optimal-homotopies-over-a-spiked-plane-with-polygonal-boundary:2017,
 title = {Computing Optimal Homotopies over a Spiked Plane with Polygonal Boundary},
 author = {B. Burton and E.W. Chambers and M.J. Van Kreveld and W. Meulemans and T.A.E. Ophelders and B. Speckmann},
 year = {2017},
 bookTitle = {Proc. 25th European Symposium on Algorithms (ESA)},
}
```
[ PDF ]
Computing Representative Networks for Braided Rivers
M. Kleinhans, M.J. van Kreveld, T.A.E. Ophelders, W.M. Sonke, B. Speckmann, and K.A.B. Verbeek.
33rd International Symposium on Computational Geometry, SoCG 2017, pp. 1—16, 2017.

－ Abstract

Drainage networks on terrains have been studied extensively from an algorithmic perspective. However, in drainage networks water flow cannot bifurcate and hence they do not model braided rivers (multiple channels which split and join, separated by sediment bars). We initiate the algorithmic study of braided rivers by employing the descending quasi Morse-Smale complex on the river bed (a polyhedral terrain), and extending it with a certain ordering of bars from the one river bank to the other. This allows us to compute a graph that models a representative channel network, consisting of lowest paths. To ensure that channels in this network are sufficiently different we define a sand function that represents the volume of sediment separating them. We show that in general the problem of computing a maximum network of non-crossing channels which are δ-different from each other (as measured by the sand function) is NP-hard. However, using our ordering between the river banks, we can compute a maximum δ-different network that respects this order in polynomial time. We implemented our approach and applied it to simulated and real-world braided rivers.
－ BibTeX
```
@article{computing-representative-networks-for-braided-rivers:2017,
    title = {Computing Representative Networks for Braided Rivers},
    author = {M. Kleinhans and M.J. van Kreveld and T.A.E. Ophelders and W.M. Sonke and B. Speckmann and K.A.B. Verbeek},
    year = {2017},
    bookTitle = {33rd International Symposium on Computational Geometry, SoCG 2017},
}
```
[ PDF ]
Computing the Fréchet Distance Between Real-Valued Surfaces
K. Buchin, T. Ophelders, and B. Speckmann.
Proc. 28th Annual Symposium on Discrete Algorithms (SODA), pp. 2443—2455, 2017.

－ Abstract

The Fréchet distance is a well-studied measure for the similarity of shapes. While efficient algorithms for computing the Fréchet distance between curves exist, there are only few results on the Fréchet distance between surfaces. Recent work has shown that the Fréchet distance is computable between piecewise linear functions f and g : M → Rk with M a triangulated surface of genus zero. We focus on the case k = 1 and M being a topological sphere or disk with constant boundary. Intuitively, we measure the distance between terrains based solely on the height function. Our main result is that in this case computing the Fréchet distance between f and g is in NP. We additionally show that already for k = 1, computing a factor 2 - ϵ approximation of the Fréchet distance is NP-hard, showing that this problem is in fact NP-complete. We also define an intermediate distance, between contour trees, which we also show to be NP-complete to compute. Finally, we discuss how our and other distance measures between contour trees relate to each other.
－ BibTeX
```
@article{computing-the-fr-chet-distance-between-real-valued-surfaces:2017,
 title = {Computing the Fréchet Distance Between Real-Valued Surfaces},
 author = {K. Buchin and T. Ophelders and B. Speckmann},
 year = {2017},
 bookTitle = {Proc. 28th Annual Symposium on Discrete Algorithms (SODA)},
}
```
[ PDF ]
GlottoVis: Visualizing Language Endangerment and Documentation
T.H.A. Castermans, B. Speckmann, K.A.B. Verbeek, M.A. Westenberg, and H. Hammarström.
Proc. 2nd Workshop on Visualization for the Digital Humanities (VIS4DH), pp. 1—5, 2017.

－ Abstract

We present GlottoVis, a system designed to visualize language endangerment as collected by UNESCO and descriptive status as collected by the Glottolog project. Glottolog records bibliographic data for the world’s (lesser known) languages. Languages are documented with increasing detail, but the number of native speakers of minority languages dwindles as their population shifts to other more dominant languages. Hence one needs to visualize documentation level and endangerment status at the same time, to browse for the most urgent cases (little documentation and high endangerment) and to direct funding aimed at describing endangered languages. GlottoVis visualizes these two properties of languages and provides an interface to search and filter. Our tool is web-based and is comprised of glyphs on a zoomable geographic map. Clustering and (visual) data aggregation are performed at each zoom level to avoid overlap of glyphs. This reduces clutter and improves readability of the visualization. Preliminary tests with expert users confirm that our tool supports their desired workflow well.
－ BibTeX
```
@article{glottovis-visualizing-language-endangerment-and-documentation:2017,
    title = {GlottoVis: Visualizing Language Endangerment and Documentation},
    author = {T.H.A. Castermans and B. Speckmann and K.A.B. Verbeek and M.A. Westenberg and H. Hammarström},
    year = {2017},
    bookTitle = {Proc. 2nd Workshop on Visualization for the Digital Humanities (VIS4DH)},
}
```
[ PDF ]
Non-Crossing Geometric Steiner Arborescences
I. Kostitsyna, B. Speckmann, and K.A.B. Verbeek.
28th International Symposium on Algorithms and Computation, ISAAC 2017, pp. 1—13, 2017.

－ Abstract

Motivated by the question of simultaneous embedding of several flow maps, we consider the problem of drawing multiple geometric Steiner arborescences with no crossings in the rectilinear and in the angle-restricted setting. When terminal-to-root paths are allowed to turn freely, we show that two rectilinear Steiner arborescences have a non-crossing drawing if neither tree necessarily completely disconnects the other tree and if the roots of both trees are "free". If the roots are not free, then we can reduce the decision problem to 2SAT. If terminal-to-root paths are allowed to turn only at Steiner points, then it is NP-hard to decide whether multiple rectilinear Steiner arborescences have a non-crossing drawing. The setting of angle-restricted Steiner arborescences is more subtle than the rectilinear case. Our NP-hardness result extends, but testing whether there exists a non-crossing drawing if the roots of both trees are free requires additional conditions to be fulfilled.
－ BibTeX
```
@article{non-crossing-geometric-steiner-arborescences:2017,
    title = {Non-Crossing Geometric Steiner Arborescences},
    author = {I. Kostitsyna and B. Speckmann and K.A.B. Verbeek},
    year = {2017},
    bookTitle = {28th International Symposium on Algorithms and Computation, ISAAC 2017},
}
```
[ PDF ]

Computing Representative Networks for Braided Rivers
M. Kleinhans, M.J. van Kreveld, T.A.E. Ophelders, W.M. Sonke, B. Speckmann, and K.A.B. Verbeek.
pp. 45—48, 2017.

－ BibTeX

@article{computing-representative-networks-for-braided-rivers:2017,
    title = {Computing Representative Networks for Braided Rivers},
    author = {M. Kleinhans and M.J. van Kreveld and T.A.E. Ophelders and W.M. Sonke and B. Speckmann and K.A.B. Verbeek},
    year = {2017},
    bookTitle = {},
}

[ PDF ]

Fréchet Isotopies to Monotone Curves
K.A. Buchin, E.W. Chambers, T.A.E. Ophelders, and B. Speckmann.
pp. 41—44, 2017.

－ BibTeX

@article{fr-chet-isotopies-to-monotone-curves:2017,
    title = {Fréchet Isotopies to Monotone Curves},
    author = {K.A. Buchin and E.W. Chambers and T.A.E. Ophelders and B. Speckmann},
    year = {2017},
    bookTitle = {},
}

[ PDF ]

Non-Crossing Drawings of Multiple Geometric Steiner Arborescences
I. Kostitsyna, B. Speckmann, and K.A.B. Verbeek.
pp. 133—136, 2017.

－ BibTeX

@article{non-crossing-drawings-of-multiple-geometric-steiner-arborescences:2017,
    title = {Non-Crossing Drawings of Multiple Geometric Steiner Arborescences},
    author = {I. Kostitsyna and B. Speckmann and K.A.B. Verbeek},
    year = {2017},
    bookTitle = {},
}

[ PDF ]

2016

Area-Preserving Simplification and Schematization of Polygonal Subdivisions
Kevin Buchin, Wouter Meulemans, André Van Renssen, and Bettina Speckmann.
ACM Transactions on Spatial Algorithms and Systems , 2(1):1—36, 2016.

－ Abstract

In this article, we study automated simplification and schematization of territorial outlines. We present a quadratic-time simplification algorithm based on an operation called edge-move. We prove that the number of edges of any nonconvex simple polygon can be reduced with this operation. Moreover, edge-moves preserve area and topology and do not introduce new orientations. The latter property in particular makes the algorithm highly suitable for schematization in which all resulting lines are required to be parallel to one of a given set of lines (orientations). To obtain such a result, we need only to preprocess the input to use only lines that are parallel to one of the given set. We present an algorithm to enforce such orientation restrictions, again without changing area or topology. Experiments show that our algorithms obtain results of high visual quality.
－ BibTeX
```
@article{area-preserving-simplification-and-schematization-of-polygonal-subdivisions:2016,
 title = {Area-Preserving Simplification and Schematization of Polygonal Subdivisions},
 author = {Kevin Buchin and Wouter Meulemans and André Van Renssen and Bettina Speckmann},
 year = {2016},
 bookTitle = {ACM Transactions on Spatial Algorithms and Systems },
}
```
[ PDF ]
[ PDF ]
Convex-Arc Drawings of Pseudolines
D. Eppstein, M. van Garderen, B. Speckmann, and T. Ueckerdt.
arXiv, 2016.

－ Abstract

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.
－ BibTeX
```
@article{convex-arc-drawings-of-pseudolines:2016,
    title = {Convex-Arc Drawings of Pseudolines},
    author = {D. Eppstein and M. van Garderen and B. Speckmann and T. Ueckerdt},
    year = {2016},
    bookTitle = {arXiv},
}
```
[ PDF ]
Distance-Sensitive Planar Point Location
B. Aronov, M.T. de Berg, D. Eppstein, M.J.M. Roeloffzen, and B. Speckmann.
Computational Geometry, 54:17–31, 2016.

－ Abstract

Let SS be a connected planar polygonal subdivision with n edges that we want to preprocess for point-location queries, and where we are given the probability γiγi that the query point lies in a polygon PiPi of SS. We show how to preprocess SS such that the query time for a point p∈Pip∈Pi depends on γiγi and, in addition, on the distance from p to the boundary of PiPi—the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time View the MathML sourceO(min⁡(log⁡n,1+log⁡area(Pi)γiΔp2)), where ΔpΔp is the shortest Euclidean distance of the query point p to the boundary of PiPi. Our structure uses O(n)O(n) space and O(nlog⁡n)O(nlog⁡n) preprocessing time. It is based on a decomposition of the regions of SS into convex quadrilaterals and triangles with the following property: for any point p∈Pip∈Pi, the quadrilateral or triangle containing p has area View the MathML sourceΩ(Δp2). For the special case where SS is a subdivision of the unit square and γi=area(Pi)γi=area(Pi), we present a simpler solution that achieves a query time of View the MathML sourceO(min⁡(log⁡n,log⁡1Δp2)). The latter solution can be extended to convex subdivisions in three dimensions.
－ BibTeX
```
@article{distance-sensitive-planar-point-location:2016,
    title = {Distance-Sensitive Planar Point Location},
    author = {B. Aronov and M.T. de Berg and D. Eppstein and M.J.M. Roeloffzen and B. Speckmann},
    year = {2016},
    bookTitle = {Computational Geometry},
}
```
[ PDF ]
[ PDF ]
Distance-Sensitive Planar Point Location
B. Aronov, M.T. de Berg, D. Eppstein, M.J.M. Roeloffzen, and B. Speckmann.
arXiv, 2016.

－ Abstract

Let S be a connected planar polygonal subdivision with n edges that we want to preprocess for point-location queries, and where we are given the probability γ i that the query point lies in a polygon P i of S . We show how to preprocess S such that the query time for a point~p∈P i depends on~γ i and, in addition, on the distance from p to the boundary of~P i ---the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time O(min(logn,1+logarea(P i )γ i Δ 2 p )) , where Δ p is the shortest Euclidean distance of the query point~p to the boundary of P i . Our structure uses O(n) space and O(nlogn) preprocessing time. It is based on a decomposition of the regions of S into convex quadrilaterals and triangles with the following property: for any point p∈P i , the quadrilateral or triangle containing~p has area Ω(Δ 2 p ) . For the special case where S is a subdivision of the unit square and γ i =area(P i ) , we present a simpler solution that achieves a query time of O(min(logn,log1Δ 2 p )) . The latter solution can be extended to convex subdivisions in three dimensions.
－ BibTeX
```
@article{distance-sensitive-planar-point-location:2016,
    title = {Distance-Sensitive Planar Point Location},
    author = {B. Aronov and M.T. de Berg and D. Eppstein and M.J.M. Roeloffzen and B. Speckmann},
    year = {2016},
    bookTitle = {arXiv},
}
```
[ PDF ]
Strict Confluent Drawing
D. Eppstein, D. Holten, M. Löffler, M. Nöllenburg, B. Speckmann, and K.A.B. Verbeek.
Journal of Computational Geometry, 7(1):22—46, 2016.

－ Abstract

We define strict confluent drawing, a form of confluent drawing in which the existence of an edge is indicated by the presence of a smooth path through a system of arcs and junctions (without crossings), and in which such a path, if it exists, must be unique. We prove that it is NP-complete to determine whether a given graph has a strict confluent drawing but polynomial to determine whether it has an outerplanar strict confluent drawing with a fixed vertex ordering (a drawing within a disk, with the vertices placed in a given order on the boundary). 
－ BibTeX
```
@article{strict-confluent-drawing:2016,
 title = {Strict Confluent Drawing},
 author = {D. Eppstein and D. Holten and M. Löffler and M. Nöllenburg and B. Speckmann and K.A.B. Verbeek},
 year = {2016},
 bookTitle = {Journal of Computational Geometry},
}
```
[ PDF ]
Visual Encoding of Dissimilarity Data Via Topology-Preserving Map Deformation
Q.W. Bouts, T. Dwyer, J. Dykes, B. Speckmann, S. Goodwin, N.H. Riche, S. Carpendale, and A. Liebman.
IEEE Transactions on Visualization and Computer Graphics, 22(9):2200—2213, 2016.

－ Abstract

We present an efficient technique for topology-preserving map deformation and apply it to the visualization of dissimilarity data in a geographic context. Map deformation techniques such as value-by-area cartograms are well studied. However, using deformation to highlight (dis)similarity between locations on a map in terms of their underlying data attributes is novel. We also identify an alternative way to represent dissimilarities on a map through the use of visual overlays. These overlays are complementary to deformation techniques and enable us to assess the quality of the deformation as well as to explore the design space of blending the two methods. Finally, we demonstrate how these techniques can be useful in several-quite different-applied contexts: travel-time visualization, social demographics research and understanding energy flowing in a wide-area power-grid.
－ BibTeX
```
@article{visual-encoding-of-dissimilarity-data-via-topology-preserving-map-deformation:2016,
 title = {Visual Encoding of Dissimilarity Data Via Topology-Preserving Map Deformation},
 author = {Q.W. Bouts and T. Dwyer and J. Dykes and B. Speckmann and S. Goodwin and N.H. Riche and S. Carpendale and A. Liebman},
 year = {2016},
 bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
}
```
[ PDF ]
An Improved Lower Bound on the Minimum Number of Triangulations
O. Aichholzer, V. Alvarez, T. Hackl, A. Pilz, B. Speckmann, and B. Vogtenhuber.
32nd International Symposium on Computational Geometry, SoCG 2016, 14-17 June 2016, Boston, Massachusetts, pp. 7.1—7.16, 2016.

－ Abstract

Upper and lower bounds for the number of geometric graphs of specific types on a given set of points in the plane have been intensively studied in recent years. For most classes of geometric graphs it is now known that point sets in convex position minimize their number. However, it is still unclear which point sets minimize the number of geometric triangulations; the so-called double circles are conjectured to be the minimizing sets. In this paper we prove that any set of n points in general position in the plane has at least Ω(2.631n) geometric triangulations. Our result improves the previously best general lower bound of Ω(2.43n) and also covers the previously best lower bound of Ω(2.63n) for a fixed number of extreme points. We achieve our bound by showing and combining several new results, which are of independent interest: 1. Adding a point on the second convex layer of a given point set (of 7 or more points) at least doubles the number of triangulations. 2. Generalized configurations of points that minimize the number of triangulations have at most ⌊n/2⌋ points on their convex hull. 3. We provide tight lower bounds for the number of triangulations of point sets with up to 15 points. These bounds further support the double circle conjecture.
－ BibTeX
```
@article{an-improved-lower-bound-on-the-minimum-number-of-triangulations:2016,
 title = {An Improved Lower Bound on the Minimum Number of Triangulations},
 author = {O. Aichholzer and V. Alvarez and T. Hackl and A. Pilz and B. Speckmann and B. Vogtenhuber},
 year = {2016},
 bookTitle = {32nd International Symposium on Computational Geometry, SoCG 2016, 14-17 June 2016, Boston, Massachusetts},
}
```
Circles in the Water: Towards Island Group Labeling
A.I. van Goethem, M.J. van Kreveld, and B. Speckmann.
Proc. of the 9th International Conference on Geographic Information Science (GIScience), pp. 293—307, 2016.

－ Abstract

Many algorithmic results are known for automated label placement on maps. However, algorithms to compute labels for groups of features, such as island groups, are largely missing. In this paper we address this issue by presenting new, efficient algorithms for island label placement in various settings. We consider straight-line and circular-arc labels that may or may not overlap a given set of islands. We concentrate on computing the line or circle that minimizes the maximum distance to the islands, measured by the closest distance. We experimentally test whether the generated labels are reasonable for various real-world island groups, and compare different options. The results are positive and validate our geometric formalizations.
－ BibTeX
```
@article{circles-in-the-water-towards-island-group-labeling:2016,
    title = {Circles in the Water: Towards Island Group Labeling},
    author = {A.I. van Goethem and M.J. van Kreveld and B. Speckmann},
    year = {2016},
    bookTitle = {Proc. of the 9th International Conference on Geographic Information Science (GIScience)},
}
```
[ PDF ]
Geo Word Clouds
K.A. Buchin, D.J.A. Creemers, A. Lazzarotto , B. Speckmann, and J.J.H.M. Wulms.
2016 IEEE Pacific Visualization Symposium (PacificVis), 19-22 April 2016, Taipei, Taiwan , pp. 144—151, 2016.

－ Abstract

Word clouds are a popular method to visualize the frequency of words in textual data. Nowadays many text-based data sets, such as Flickr tags, are geo-referenced, that is, they have an important spatial component. However, existing automated methods to generate word clouds are unable to incorporate such spatial information. We introduce geo word clouds: word clouds which capture not only the frequency but also the spatial relevance of words. Our input is a set of locations from one (or more) geographic regions with (possibly several) text labels per location. We aggregate word frequencies according to point clusters and employ a greedy strategy to place appropriately sized labels without overlap as close as possible to their corresponding locations. While doing so we "draw" the spatial shapes of the geographic regions with the corresponding labels. We experimentally explore trade-offs concerning the location of labels, their relative sizes and the number of spatial clusters. The resulting word clouds are visually pleasing and have a low error in terms of relative scaling and locational accuracy of words, while using a small number of clusters per label.
－ BibTeX
```
@article{geo-word-clouds:2016,
    title = {Geo Word Clouds},
    author = {K.A. Buchin and D.J.A. Creemers and A. Lazzarotto  and B. Speckmann and J.J.H.M. Wulms},
    year = {2016},
    bookTitle = {2016 IEEE Pacific Visualization Symposium (PacificVis), 19-22 April 2016, Taipei, Taiwan },
}
```
[ PDF ]
GlamMap: Geovisualization for e-Humanities
T.H.A. Castermans, B. Speckmann, K.A.B. Verbeek, M.A. Westenberg, A. Betti, and H. van den Berg.
Proc. 1st Workshop on Visualization for the Digital Humanities (VIS4DH), pp. 1—5, 2016.

－ Abstract

This paper presents GlamMap, a visualization tool for large, multi-variate georeferenced humanities data sets. Our approach visualizes the data as glyphs on a zoomable geographic map, and performs clustering and data aggregation at each zoom level to avoid clutter and to prevent overlap of symbols. GlamMap was developed for the Galleries, Libraries, Archives, and Museums (GLAM) domain in cooperation with researchers in philosophy. We demonstrate the usefulness of our approach by a case study on history of logic, which involves navigation and exploration of 7100 bibliographic records, and scalability on a data set of sixty million book records.
－ BibTeX
```
@article{glammap-geovisualization-for-e-humanities:2016,
    title = {GlamMap: Geovisualization for e-Humanities},
    author = {T.H.A. Castermans and B. Speckmann and K.A.B. Verbeek and M.A. Westenberg and A. Betti and H. van den Berg},
    year = {2016},
    bookTitle = {Proc. 1st Workshop on Visualization for the Digital Humanities (VIS4DH)},
}
```
[ PDF ]
GlamMapping Trove
A. Betti, T.H.A. Castermans, B. Speckmann, H. van den Berg, and K.A.B. Verbeek.
Proc. VALA, 2016.

－ Abstract

This paper presents the current state of development of GlamMap, a visualisation tool that displays library metadata on an interactive, computer-generated geographic map. The focus in the paper is on the most crucial improvement achieved in the development of the tool: GlamMapping Trove. The visualisation of Trove’s sixty-million book records is possible thanks to an improved database structure, more efficient data retrieval, and more scalable visualisation algorithms. The paper analyses problems encountered in visualising massive datasets, describes remaining challenges for the tool, and presents a use case demonstrating GlamMap’s ability to serve researchers in the history of ideas.
－ BibTeX
```
@article{glammapping-trove:2016,
    title = {GlamMapping Trove},
    author = {A. Betti and T.H.A. Castermans and B. Speckmann and H. van den Berg and K.A.B. Verbeek},
    year = {2016},
    bookTitle = {Proc. VALA},
}
```
[ PDF ]
Grouping Time-Varying Data for Interactive Exploration
A.I. van Goethem, M.J. van Kreveld, M. Löffler, B. Speckmann, and F. Staals.
Proc. of the 32nd annual Symposium on Computational Geometry (SoCG), pp. 1—16, 2016.

－ Abstract

We present algorithms and data structures that support the interactive analysis of the grouping structure of one-, two-, or higher-dimensional time-varying data while varying all defining parameters. Grouping structures characterise important patterns in the temporal evaluation of sets of time-varying data. We follow Buchin et al. [JoCG 2015] who define groups using three parameters: group-size, group-duration, and inter-entity distance. We give upper and lower bounds on the number of maximal groups over all parameter values, and show how to compute them efficiently. Furthermore, we describe data structures that can report changes in the set of maximal groups in an output-sensitive manner. Our results hold in R^d for fixed d.
－ BibTeX
```
@article{grouping-time-varying-data-for-interactive-exploration:2016,
    title = {Grouping Time-Varying Data for Interactive Exploration},
    author = {A.I. van Goethem and M.J. van Kreveld and M. Löffler and B. Speckmann and F. Staals},
    year = {2016},
    bookTitle = {Proc. of the 32nd annual Symposium on Computational Geometry (SoCG)},
}
```
[ PDF ]
Modeling Checkpoint-Based Movement with the Earth Mover’s Distance
M. Duckham, M.J. van Kreveld, R. Purves, B. Speckmann, Y. Tao, K.A.B. Verbeek, and J. Wood.
Geographic Information Science , pp. 225—239, 2016.

－ Abstract

Movement data comes in various forms, including trajectory data and checkpoint data. While trajectories give detailed information about the movement of individual entities, checkpoint data in its simplest form does not give identities, just counts at checkpoints. However, checkpoint data is of increasing interest since it is readily available due to privacy reasons and as a by-product of other data collection. In this paper we propose to use the Earth Mover’s Distance as a versatile tool to reconstruct individual movements or flow based on checkpoint counts at different times. We analyze the modeling possibilities and provide experiments that validate model predictions, based on coarse-grained aggregations of data about actual movements of couriers in London, UK. While we cannot expect to reconstruct precise individual movements from highly granular checkpoint data, the evaluation does show that the approach can generate meaningful estimates of object movements.
－ BibTeX
```
@article{modeling-checkpoint-based-movement-with-the-earth-mover-s-distance:2016,
 title = {Modeling Checkpoint-Based Movement with the Earth Mover’s Distance},
 author = {M. Duckham and M.J. van Kreveld and R. Purves and B. Speckmann and Y. Tao and K.A.B. Verbeek and J. Wood},
 year = {2016},
 bookTitle = {Geographic Information Science },
}
```
[ PDF ]

Computing the Fréchet Distance Between Real-Valued Surfaces
K.A. Buchin, T.A.E. Ophelders, and B. Speckmann.
pp. 239—242, 2016.

－ BibTeX

@article{computing-the-fr-chet-distance-between-real-valued-surfaces:2016,
    title = {Computing the Fréchet Distance Between Real-Valued Surfaces},
    author = {K.A. Buchin and T.A.E. Ophelders and B. Speckmann},
    year = {2016},
    bookTitle = {},
}

[ PDF ]

Grouping Time-Varying Data for Interactive Exploration
A.I. van Goethem, M.J. van Kreveld, M. Löffler, B. Speckmann, and F. Staals.
pp. 7—10, 2016.

－ BibTeX

@article{grouping-time-varying-data-for-interactive-exploration:2016,
    title = {Grouping Time-Varying Data for Interactive Exploration},
    author = {A.I. van Goethem and M.J. van Kreveld and M. Löffler and B. Speckmann and F. Staals},
    year = {2016},
    bookTitle = {},
}

[ PDF ]

2015

A Formal Approach to the Automated Labeling of Groups of Features
A. Reimer, A.I. Goethem, van, M. Rylov, M.J. Kreveld, van, and B. Speckmann.
Cartography and Geographic Information Science, 42(4):333—344, 2015.

－ Abstract

A variety of recurring geographic entities form collections of point, line, or area features. Examples are groups of islands (Archipelago), relief features in deserts, periglacial lakes or geomorphological forms, such as drumlins and sinkholes. All these groups of features might be best identified with a single label. Surprisingly, the (automated) labeling of groups of features has received little attention so far. We propose a framework to determine formal measures that describe the geometric aspects of the cartographic design space for labeling feature groups. Our framework gives rise to a large variety of geometrically optimal labels. We list the optimal label positions and shapes for which labels can already be computed using existing algorithms. However, in many of the cases computing optimal label placements is still an open algorithmic question which can readily be investigated in future work. Once the necessary algorithms are developed, our framework provides an objective basis to investigate the geometric measures used by cartographers to label groups of features. We preliminarily explore the applicability of our framework using one of the geometric optimality choices. Keywords: automated cartography, labeling, groups of features
－ BibTeX
```
@article{a-formal-approach-to-the-automated-labeling-of-groups-of-features:2015,
    title = {A Formal Approach to the Automated Labeling of Groups of Features},
    author = {A. Reimer and A.I. Goethem, van and M. Rylov and M.J. Kreveld, van and B. Speckmann},
    year = {2015},
    bookTitle = {Cartography and Geographic Information Science},
}
```
[ PDF ]
Algorithms for Necklace Maps
B. Speckmann and K.A.B. Verbeek.
International Journal of Computational Geometry and Applications, 25(1):15—36, 2015.

－ Abstract

Necklace maps visualize quantitative data associated with regions by placing scaled symbols, usually disks, without overlap on a closed curve (the necklace) surrounding the map regions. Each region is projected onto an interval on the necklace that contains its symbol. In this paper we address the algorithmic question how to maximize symbol sizes while keeping symbols disjoint and inside their intervals. For that we reduce the problem to a one-dimensional problem which we solve efficiently. Solutions to the one-dimensional problem provide a very good approximation for the original necklace map problem. We consider two variants: Fixed-Order, where an order for the symbols on the necklace is given, and Any-Order where any symbol order is possible. The Fixed-Order problem can be solved in O(n log n) time. We show that the Any-Order problem is NP-hard for certain types of intervals and give an exact algorithm for the decision version. This algorithm is fixed-parameter tractable in the thickness K of the input. Our algorithm runs in O(n log n + n2K4K) time which can be improved to O(n log n + nK2K) time using a heuristic. We implemented our algorithm and evaluated it experimentally. Keywords: Necklace maps; scheduling; automated cartography
－ BibTeX
```
@article{algorithms-for-necklace-maps:2015,
    title = {Algorithms for Necklace Maps},
    author = {B. Speckmann and K.A.B. Verbeek},
    year = {2015},
    bookTitle = {International Journal of Computational Geometry and Applications},
}
```
[ PDF ]
Angle-Restricted Steiner Arborescences for Flow Map Layout
K. Buchin, B. Speckmann, and K.A.B. Verbeek.
Algorithmica, 72(2):656—685, 2015.

－ Abstract

We introduce a new variant of the geometric Steiner arborescence problem, motivated by the layout of flow maps. Flow maps show the movement of objects between places. They reduce visual clutter by bundling curves smoothly and avoiding self-intersections. To capture these properties, our angle-restricted Steiner arborescences, or flux trees, connect several targets to a source with a tree of minimal length whose arcs obey a certain restriction on the angle they form with the source. We study the properties of optimal flux trees and show that they are crossing-free and consist of logarithmic spirals and straight lines. Flux trees have the shallow-light property. We show that computing optimal flux trees is NP-hard. Hence we consider a variant of flux trees which uses only logarithmic spirals. Spiral trees approximate flux trees within a factor depending on the angle restriction. Computing optimal spiral trees remains NP-hard, but we present an efficient 2-approximation, which can be extended to avoid "positive monotone" obstacles. Keywords: Steiner arborescences; Flow maps; Computational geometry; Automated cartography
－ BibTeX
```
@article{angle-restricted-steiner-arborescences-for-flow-map-layout:2015,
    title = {Angle-Restricted Steiner Arborescences for Flow Map Layout},
    author = {K. Buchin and B. Speckmann and K.A.B. Verbeek},
    year = {2015},
    bookTitle = {Algorithmica},
}
```
[ PDF ]
Exploring Curved Schematization of Territorial Outlines
A.I. Goethem, van, W. Meulemans, B. Speckmann, and J.D. Wood.
IEEE Transactions on Visualization and Computer Graphics, 21(8):889—902, 2015.

－ Abstract

Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization; most previous work focuses on straight lines. We present a new algorithm for areapreserving curved schematization of territorial outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs.We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of "curviness": we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. To evaluate the effectiveness of our design choices, we present a geometric evaluation of the resulting schematizations. Besides the geometric qualities of our algorithm, we also investigate the potential of curved schematization as a concept. We conducted an online user study investigating the effectiveness of curved schematizations compared to straight-line schematizations. While the visual complexity of curved shapes was judged higher than that of straight-line shapes, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents. Keywords: Schematization; algorithm; circular arcs; user study
－ BibTeX
```
@article{exploring-curved-schematization-of-territorial-outlines:2015,
    title = {Exploring Curved Schematization of Territorial Outlines},
    author = {A.I. Goethem, van and W. Meulemans and B. Speckmann and J.D. Wood},
    year = {2015},
    bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
}
```
[ PDF ]
[ PDF ]
Improved Grid Map Layout by Point Set Matching
D. Eppstein, M.J. Kreveld, van, B. Speckmann, and F. Staals.
International Journal of Computational Geometry and Applications, 25(2):101—122, 2015.

－ Abstract

Associating the regions of a geographic subdivision with the cells of a grid is a basic operation that is used in various types of maps, like spatially ordered treemaps and Origin-Destination maps (OD maps). In these cases the regular shapes of the grid cells allow easy representation of extra information about the regions. The main challenge is to find an association that allows a user to find a region in the grid quickly. We call the representation of a set of regions as a grid a grid map. We introduce a new approach to solve the association problem for grid maps by formulating it as a point set matching problem: Given two sets A (the centroids of the regions) and B (the grid centres) of n points in the plane, compute an optimal one-to-one matching between A and B. We identify three optimisation criteria that are important for grid map layout: maximise the number of adjacencies in the grid that are also adjacencies of the regions, minimise the sum of the distances between matched points, and maximise the number of pairs of points in A for which the matching preserves the directional relation (SW, NW, etc.). We consider matchings that minimise the L1-distance (Manhattan-distance), the ranked L1-distance, and the L22-distance, since one can expect that minimising distances implicitly helps to fulfill the other criteria. We present algorithms to compute such matchings and perform an experimental comparison that also includes a previous method to compute a grid map. The experiments show that our more global, matching-based algorithm outperforms previous, more local approaches with respect to all three optimisation criteria. Keywords: Grid map; point-set matching; visualization
－ BibTeX
```
@article{improved-grid-map-layout-by-point-set-matching:2015,
    title = {Improved Grid Map Layout by Point Set Matching},
    author = {D. Eppstein and M.J. Kreveld, van and B. Speckmann and F. Staals},
    year = {2015},
    bookTitle = {International Journal of Computational Geometry and Applications},
}
```
[ PDF ]
Mosaic Drawings and Cartograms
R.G. Cano, K. Buchin, T.H.A. Castermans, A. Pieterse, W.M. Sonke, and B. Speckmann.
Computer Graphics Forum, 34(3):361—370, 2015.

－ Abstract

Cartograms visualize quantitative data about a set of regions such as countries or states. There are several different types of cartograms and – for some – algorithms to automatically construct them exist. We focus on mosaic cartograms: cartograms that use multiples of simple tiles – usually squares or hexagons – to represent regions. Mosaic cartograms communicate well data that consist of, or can be cast into, small integer units (for example, electorial college votes). In addition, they allow users to accurately compare regions and can often maintain a (schematized) version of the input regions’ shapes. We propose the first fully automated method to construct mosaic cartograms. To do so, we first introduce mosaic drawings of triangulated planar graphs. We then show how to modify mosaic drawings into mosaic cartograms with low cartographic error while maintaining correct adjacencies between regions. We validate our approach experimentally and compare to other cartogram methods.
－ BibTeX
```
@article{mosaic-drawings-and-cartograms:2015,
    title = {Mosaic Drawings and Cartograms},
    author = {R.G. Cano and K. Buchin and T.H.A. Castermans and A. Pieterse and W.M. Sonke and B. Speckmann},
    year = {2015},
    bookTitle = {Computer Graphics Forum},
}
```
[ PDF ]
Trajectory Grouping Structure
K.A. Buchin, M.E. Buchin, M.J. van Kreveld, B. Speckmann, and F. Staals.
Journal of Computational Geometry, 6(1):75—98, 2015.

－ Abstract

The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept from topology. The trajectory grouping structure has three natural parameters that allow more global views of the data in group size, group duration, and entity inter-distance. We prove complexity bounds on the maximum number of maximal groups that can be present, and give algorithms to compute the grouping structure efficiently. We also study how the trajectory grouping structure can be made robust, that is, how brief interruptions of groups can be disregarded in the global structure, adding a notion of persistence to the structure. Furthermore, we showcase the results of experiments using data generated by the NetLogo flocking model and from the Starkey project. The Starkey data describe the movement of elk, deer, and cattle. Although there is no ground truth for the grouping structure in this data, the experiments show that the trajectory grouping structure is plausible and has the desired effects when changing the essential parameters. Our research provides the first complete study of trajectory group evolvement, including combinatorial, algorithmic, and experimental results. 
－ BibTeX
```
@article{trajectory-grouping-structure:2015,
 title = {Trajectory Grouping Structure},
 author = {K.A. Buchin and M.E. Buchin and M.J. van Kreveld and B. Speckmann and F. Staals},
 year = {2015},
 bookTitle = {Journal of Computational Geometry},
}
```
[ PDF ]
Clustered Edge Routing
Q.W. Bouts and B. Speckmann.
2015 IEEE Pacific Visualization Symposium (PacificVis2015, Hangzhou, China, April 14-17, 2015), pp. 55—62, 2015.

－ Abstract

The classic method to depict graphs is a node-link diagram where vertices (nodes) are associated with each object and edges (links) connect related objects. However, node-link diagrams quickly appear cluttered and unclear, even for moderately sized graphs. If the positions of the nodes are fixed then suitable link routing is the only option to reduce clutter. We present a novel link clustering and routing algorithm which respects (and if desired refines) user-defined clusters on links. If no clusters are defined a priori we cluster based on geometric criteria, that is, based on a well-separated pair decomposition (WSPD).We route link clusters individually on a sparse visibility spanner. To completely avoid ambiguity we draw each individual link and ensure that clustered links follow the same path in the routing graph. We prove that the clusters induced by the WSPD consist of compatible links according to common similarity measures as formalized by Holten and van Wijk [17]. The greedy sparsification of the visibility graph allows us to easily route around obstacles. Our experimental results are visually appealing and convey a sense of abstraction and order.
－ BibTeX
```
@article{clustered-edge-routing:2015,
    title = {Clustered Edge Routing},
    author = {Q.W. Bouts and B. Speckmann},
    year = {2015},
    bookTitle = {2015 IEEE Pacific Visualization Symposium (PacificVis2015, Hangzhou, China, April 14-17, 2015)},
}
```
[ PDF ]
Computing the Similarity Between Moving Curves
K. Buchin, T.A.E. Ophelders, and B. Speckmann.
Proc. 23rd Annual European Symposium on Algorithms (ESA), pp. 928—940, 2015.

－ Abstract

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fréchet distance between surfaces. While the Fréchet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality.
－ BibTeX
```
@article{computing-the-similarity-between-moving-curves:2015,
    title = {Computing the Similarity Between Moving Curves},
    author = {K. Buchin and T.A.E. Ophelders and B. Speckmann},
    year = {2015},
    bookTitle = {Proc. 23rd Annual European Symposium on Algorithms (ESA)},
}
```
[ PDF ]
Geometric K Shortest Paths
S. Eriksson-Bique, J. Hershberger, V. Polishchuk, B. Speckmann, S. Suri, T. Talvitie, K.A.B. Verbeek, and H. Yildiz.
Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'15, San Diego CA, USA, January 4-6, 2015), pp. 1616—1625, 2015.

－ Abstract

We consider the problem of computing k shortest paths in a two-dimensional environment with polygonal obstacles, where the jth path, for 1 = j = k, is the shortest path in the free space that is also homotopically distinct from each of the first j – 1 paths. In fact, we consider a more general problem: given a source point s, construct a partition of the free space, called the kth shortest path map (k-SPM), in which the homotopy of the kth shortest path in a region has the same structure. Our main combinatorial result establishes a tight bound of T(k2h + kn) on the worst-case complexity of this map. We also describe an O((k3h + k2n) log (kn)) time algorithm for constructing the map. In fact, the algorithm constructs the jth map for every j = k. Finally, we present a simple visibility-based algorithm for computing the k shortest paths between two fixed points. This algorithm runs in O(m log n + k) time and uses O(m + k) space, where m is the size of the visibility graph. This latter algorithm can be extended to compute k shortest simple (non-self-intersecting) paths, taking O(k2 m(m + kn) log (kn)) time. We invite the reader to play with our applet demonstrating k-SPMs [10].
－ BibTeX
```
@article{geometric-k-shortest-paths:2015,
    title = {Geometric K Shortest Paths},
    author = {S. Eriksson-Bique and J. Hershberger and V. Polishchuk and B. Speckmann and S. Suri and T. Talvitie and K.A.B. Verbeek and H. Yildiz},
    year = {2015},
    bookTitle = {Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'15, San Diego CA, USA, January 4-6, 2015)},
}
```
[ PDF ]
Towards Characterizing Graphs with a Sliceable Rectangular Dual
V. Kusters and B. Speckmann.
Graph Drawing and Network Visualization, pp. 460—471, 2015.

－ Abstract

Let G be a plane triangulated graph. A rectangular dual of G is a partition of a rectangle R into a set R of interior-disjoint rectangles, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge. A rectangular dual is sliceable if it can be recursively subdivided along horizontal or vertical lines. A graph is rectangular if it has a rectangular dual and sliceable if it has a sliceable rectangular dual. There is a clear characterization of rectangular graphs. However, a full characterization of sliceable graphs is still lacking. The currently best result (Yeap and Sarrafzadeh, 1995) proves that all rectangular graphs without a separating 4-cycle are slice- able. In this paper we introduce a recursively defined class of graphs and prove that these graphs are precisely the nonsliceable graphs with exactly one separating 4-cycle.
－ BibTeX
```
@article{towards-characterizing-graphs-with-a-sliceable-rectangular-dual:2015,
 title = {Towards Characterizing Graphs with a Sliceable Rectangular Dual},
 author = {V. Kusters and B. Speckmann},
 year = {2015},
 bookTitle = {Graph Drawing and Network Visualization},
}
```
[ PDF ]
Trajectory Grouping Structure Under Geodesic Distance
I. Kostitsyna, M.J. van Kreveld, M. Löffler, B. Speckmann, and F. Staals.
Proc. 31st International Symposium on Computational Geometry (SoCG), pp. 674—688, 2015.

－ Abstract

In recent years trajectory data has become one of the main types of geographic data, and hence algorithmic tools to handle large quantities of trajectories are essential. A single trajectory is typically represented as a sequence of time-stamped points in the plane. In a collection of trajectories one wants to detect maximal groups of moving entities and their behaviour (merges and splits) over time. This information can be summarized in the trajectory grouping structure. Significantly extending the work of Buchin et al. [WADS 2013] into a realistic setting, we show that the trajectory grouping structure can be computed efficiently also if obstacles are present and the distance between the entities is measured by geodesic distance. We bound the number of critical events: times at which the distance between two subsets of moving entities is exactly ", where " is the threshold distance that determines whether two entities are close enough to be in one group. In case the n entities move in a simple polygon along trajectories with τ vertices each we give an O(τn2) upper bound, which is tight in the worst case. In case of well-spaced obstacles we give an O(τ (n2 + mλ4(n))) upper bound, where m is the total complexity of the obstacles, and λs(n) denotes the maximum length of a Davenport-Schinzel sequence of n symbols of order s. In case of general obstacles we give an O(τ min{n2 + m3λ4(n), n2m2}) upper bound. Furthermore, for all cases we provide efficient algorithms to compute the critical events, which in turn leads to efficient algorithms to compute the trajectory grouping structure.
－ BibTeX
```
@article{trajectory-grouping-structure-under-geodesic-distance:2015,
 title = {Trajectory Grouping Structure Under Geodesic Distance},
 author = {I. Kostitsyna and M.J. van Kreveld and M. Löffler and B. Speckmann and F. Staals},
 year = {2015},
 bookTitle = {Proc. 31st International Symposium on Computational Geometry (SoCG)},
}
```

Clustered Edge Routing
Q.W. Bouts and B. Speckmann.
pp. 149—152, 2015.

－ BibTeX

@article{clustered-edge-routing:2015,
    title = {Clustered Edge Routing},
    author = {Q.W. Bouts and B. Speckmann},
    year = {2015},
    bookTitle = {},
}

[ PDF ]

Computing the Similarity Between Moving Curves
K.A. Buchin, T.A.E. Ophelders, and B. Speckmann.
pp. 208—211, 2015.

－ BibTeX

@article{computing-the-similarity-between-moving-curves:2015,
    title = {Computing the Similarity Between Moving Curves},
    author = {K.A. Buchin and T.A.E. Ophelders and B. Speckmann},
    year = {2015},
    bookTitle = {},
}

[ PDF ]

Mosaic Drawings and Cartograms
R.G. Cano, K.A. Buchin, T.H.A. Castermans, A. Pieterse, W.M. Sonke, and B. Speckmann.
pp. 153—156, 2015.

－ Abstract

Cartograms visualize quantitative data about a set of regions such as countries or states. There are several different types of cartograms and – for some – algorithms to automatically construct them exist. We focus on mosaic cartograms: cartograms that use multiples of simple tiles – usually squares or hexagons – to represent regions. Mosaic cartograms communicate well data that consist of, or can be cast into, small integer units (for example, electorial college votes). In addition, they allow users to accurately compare regions and can often maintain a (schematized) version of the input regions’ shapes. We propose the first fully automated method to construct mosaic cartograms. To do so, we first introduce mosaic drawings of triangulated planar graphs. We then show how to modify mosaic drawings into mosaic cartograms with low cartographic error while maintaining correct adjacencies between regions. We validate our approach experimentally and compare to other cartogram methods.
－ BibTeX
```
@article{mosaic-drawings-and-cartograms:2015,
    title = {Mosaic Drawings and Cartograms},
    author = {R.G. Cano and K.A. Buchin and T.H.A. Castermans and A. Pieterse and W.M. Sonke and B. Speckmann},
    year = {2015},
    bookTitle = {},
}
```
[ PDF ]
Optimal Straight-Line Labels for Island Groups
A.I. van Goethem, M.J. van Kreveld, Andreas W. Reimer, M. Rylov, and B. Speckmann.
pp. 145—148, 2015.

－ Abstract

Maps are used to solve a wide variety of tasks, ranging from navigation to analysis. Often, the quality of a map is directly related to the quality of its labelling. Consequently, a lot of research has focussed on the automatization of the labelling process. Surprisingly the (automated) labelling of island groups has received little attention so far. This is at least partially caused by the lack of cartographic principles. 31 Though extensive guidelines for map labelling exist, information on the labelling of groups of islands is surprisingly sparse. We define a formal framework for island labelling. The framework spawns a large series of unexplored computational geometry problems, which are interesting for the CG-community. In this paper we start by looking at a non-overlapping, straight label. We describe two algorithms for a straight-line label that is, or is not, allowed overlap with islands. Furthermore, we discus several extensions to these algorithms solving closely related problems.
－ BibTeX
```
@article{optimal-straight-line-labels-for-island-groups:2015,
    title = {Optimal Straight-Line Labels for Island Groups},
    author = {A.I. van Goethem and M.J. van Kreveld and Andreas W. Reimer and M. Rylov and B. Speckmann},
    year = {2015},
    bookTitle = {},
}
```
[ PDF ]
Computing the Similarity Between Moving Curves
K. Buchin, T.A.E. Ophelders, and B. Speckmann.
2015.

－ Abstract

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fr\'echet distance between surfaces. While the Fr\'echet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality.
－ BibTeX
```
@article{computing-the-similarity-between-moving-curves:2015,
    title = {Computing the Similarity Between Moving Curves},
    author = {K. Buchin and T.A.E. Ophelders and B. Speckmann},
    year = {2015},
    bookTitle = {},
}
```
[ PDF ]
Analysis and Visualisation of Movement
U. Demšar, K.A. Buchin, F. Cagnacci, K. Safi, B. Speckmann, N. de Weghe, D. Weiskopf, and R. Weibel.
Movement Ecology, 3(5):, 2015.

－ Abstract

The processes that cause and influence movement are one of the main points of enquiry in movement ecology. However, ecology is not the only discipline interested in movement: a number of information sciences are specialising in analysis and visualisation of movement data. The recent explosion in availability and complexity of movement data has resulted in a call in ecology for new appropriate methods that would be able to take full advantage of the increasingly complex and growing data volume. One way in which this could be done is to form interdisciplinary collaborations between ecologists and experts from information sciences that analyse movement. In this paper we present an overview of new movement analysis and visualisation methodologies resulting from such an interdisciplinary research network: the European COST Action "MOVE - Knowledge Discovery from Moving Objects" (http://www.move-cost.info). This international network evolved over four years and brought together some 140 researchers from different disciplines: those that collect movement data (out of which the movement ecology was the largest represented group) and those that specialise in developing methods for analysis and visualisation of such data (represented in MOVE by computational geometry, geographic information science, visualisation and visual analytics). We present MOVE achievements and at the same time put them in ecological context by exploring relevant ecological themes to which MOVE studies do or potentially could contribute.
－ BibTeX
```
@article{analysis-and-visualisation-of-movement:2015,
 title = {Analysis and Visualisation of Movement},
 author = {U. Demšar and K.A. Buchin and F. Cagnacci and K. Safi and B. Speckmann and N. de Weghe and D. Weiskopf and R. Weibel},
 year = {2015},
 bookTitle = {Movement Ecology},
}
```
[ PDF ]

2014

Automatische Schematisering Met Gebogen Lijnen
A.I. Goethem, van, H.J. Haverkort, W. Meulemans, A. Reimer, B. Speckmann, and J.D. Wood.
Geo-Info, 2014(2):10—13, 2014.

－ BibTeX

@article{automatische-schematisering-met-gebogen-lijnen:2014,
    title = {Automatische Schematisering Met Gebogen Lijnen},
    author = {A.I. Goethem, van and H.J. Haverkort and W. Meulemans and A. Reimer and B. Speckmann and J.D. Wood},
    year = {2014},
    bookTitle = {Geo-Info},
}

[ PDF ]

Moving Beyond the Point: an Agenda for Research in Movement Analysis with Real Data
R.S. Purves, P. Laube, M. Buchin, and B. Speckmann.
Computers, Environment and Urban Systems, 47:1—4, 2014.

－ BibTeX

@article{moving-beyond-the-point-an-agenda-for-research-in-movement-analysis-with-real-data:2014,
    title = {Moving Beyond the Point: an Agenda for Research in Movement Analysis with Real Data},
    author = {R.S. Purves and P. Laube and M. Buchin and B. Speckmann},
    year = {2014},
    bookTitle = {Computers, Environment and Urban Systems},
}

On the Number of Regular Edge Labelings
K. Buchin, B. Speckmann, and S. Verdonschot.
Discrete Mathematics and Theoretical Computer Science, 16(3):215—228, 2014.

－ Abstract

We prove that any irreducible triangulation on n vertices has O (4:6807n ) regular edge labeling,s and that there are irreducible triangulations on n vertices with (3:0426n ) regular edge labelings. Our upper bound relies on a novel application of Shearer's entropy lemma. As an example of the wider applicability of this technique, we also improve the upper bound on the number of 2-orientations of a quadrangulation to O (1:87n ). Keywords: Counting; Regular edge labeling; Shearer's entropy lemma
－ BibTeX
```
@article{on-the-number-of-regular-edge-labelings:2014,
    title = {On the Number of Regular Edge Labelings},
    author = {K. Buchin and B. Speckmann and S. Verdonschot},
    year = {2014},
    bookTitle = {Discrete Mathematics and Theoretical Computer Science},
}
```
[ PDF ]
Plane Graphs with Parity Constraints
O. Aichholzer, T. Hackl, M. Hoffmann, A. Pilz, G. Rote, B. Speckmann, and B. Vogtenhuber.
Graphs and Combinatorics, 30(1):47—69, 2014.

－ Abstract

Let S be a set of n points in general position in the plane. Together with S we are given a set of parity constraints, that is, every point of S is labeled either even or odd. A graph G on S satisfies the parity constraint of a point p¿S if the parity of the degree of p in G matches its label. In this paper, we study how well various classes of planar graphs can satisfy arbitrary parity constraints. Specifically, we show that we can always find a plane tree, a two-connected outerplanar graph, or a pointed pseudo-triangulation that satisfy all but at most three parity constraints. For triangulations we can satisfy about 2/3 of the parity constraints and we show that in the worst case there is a linear number of constraints that cannot be fulfilled. In addition, we prove that for a given simple polygon H with polygonal holes on S, it is NP-complete to decide whether there exists a triangulation of H that satisfies all parity constraints.
－ BibTeX
```
@article{plane-graphs-with-parity-constraints:2014,
    title = {Plane Graphs with Parity Constraints},
    author = {O. Aichholzer and T. Hackl and M. Hoffmann and A. Pilz and G. Rote and B. Speckmann and B. Vogtenhuber},
    year = {2014},
    bookTitle = {Graphs and Combinatorics},
}
```
[ PDF ]
Similarity of Trajectories Taking into Account Geographic Context
M. Buchin, S. Dodge, and B. Speckmann.
Journal of Spatial Information Science, 9:101—124, 2014.

－ Abstract

The movements of animals, people, and vehicles are embedded in a geographic context. This context influences the movement and may cause the formation of certain behavioral responses. Thus, it is essential to include context parameters in the study of movement and the development of movement pattern analytics. Advances in sensor technologies and positioning devices provide valuable data not only of moving agents but also of the circumstances embedding the movement in space and time. Developing knowledge discovery methods to investigate the relation between movement and its surrounding context is a major challenge in movement analysis today. In this paper we show how to integrate geographic context into the similarity analysis of movement data. For this, we discuss models for geographic context of movement data. Based on this we develop simple but efficient context-aware similarity measures for movement trajectories, which combine a spatial and a contextual distance. These are based on well-known similarity measures for trajectories, such as the Hausdorff, Fréchet, or equal time distance. We validate our approach by applying these measures to movement data of hurricanes and albatross. Keywords: movement data, geographic context, spatiotemporal analytics, similarity measures, trajectory analysis, modeling context, environmental factors, Fréchet distance, Hausdorff distance, equal time distance
－ BibTeX
```
@article{similarity-of-trajectories-taking-into-account-geographic-context:2014,
    title = {Similarity of Trajectories Taking into Account Geographic Context},
    author = {M. Buchin and S. Dodge and B. Speckmann},
    year = {2014},
    bookTitle = {Journal of Spatial Information Science},
}
```
[ PDF ]
Stenomaps: Shorthand for Shapes
A.I. Goethem, van, A. Reimer, B. Speckmann, and J.D. Wood.
IEEE Transactions on Visualization and Computer Graphics, 20(12):2053—2062, 2014.

－ Abstract

We address some of the challenges in representing spatial data with a novel form of geometric abstraction – the stenomap. The stenomap comprises a series of smoothly curving linear glyphs that each represent both the boundary and the area of a polygon. We present an efficient algorithm to automatically generate these open, C1-continuous splines from a set of input polygons. Feature points of the input polygons are detected using the medial axis to maintain important shape properties. We use dynamic programming to compute a planar non-intersecting spline representing each polygon’s base shape. The results are stylised glyphs whose appearance may be parameterised and that offer new possibilities in the ‘cartographic design space’. We compare our glyphs with existing forms of geometric schematisation and discuss their relative merits and shortcomings. We describe several use cases including the depiction of uncertain model data in the form of hurricane track forecasting; minimal ink thematic mapping; and the depiction of continuous statistical data.
－ BibTeX
```
@article{stenomaps-shorthand-for-shapes:2014,
    title = {Stenomaps: Shorthand for Shapes},
    author = {A.I. Goethem, van and A. Reimer and B. Speckmann and J.D. Wood},
    year = {2014},
    bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
}
```
[ PDF ]
Treemaps with Bounded Aspect Ratio
M.T. Berg, de, B. Speckmann, and V. Weele, van der.
Computational Geometry, 47(6):683—693, 2014.

－ Abstract

Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree T where the weight of each node is the sum of the weights of its children. A treemap for T is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of T and its area is proportional to the weight of the corresponding node. An important quality criterion for treemaps is the aspect ratio of its regions. One cannot bound the aspect ratio if the regions are restricted to be rectangles. In contrast, polygonal partitions, that use convex polygons, can have bounded aspect ratio. We are the first to obtain convex partitions with optimal aspect ratio O(depth(T)). However, depth(T) still depends on the input tree. Hence we introduce a new type of treemaps, namely orthoconvex treemaps, where regions representing leaves are rectangles, L-, and S-shapes, and regions representing internal nodes are orthoconvex polygons. We prove that any input tree, irrespective of the weights of the nodes and the depth of the tree, admits an orthoconvex treemap of constant aspect ratio. We also obtain several specialized results for single-level treemaps, that is, treemaps where the input tree has depth 1. Keywords: Convex treemap; Orthoconvex treemap; Aspect ratio
－ BibTeX
```
@article{treemaps-with-bounded-aspect-ratio:2014,
    title = {Treemaps with Bounded Aspect Ratio},
    author = {M.T. Berg, de and B. Speckmann and V. Weele, van der},
    year = {2014},
    bookTitle = {Computational Geometry},
}
```
[ PDF ]
Triangulating and Guarding Realistic Polygons
G. Aloupis, P. Bose, V. Dujmovic, C.M. Gray, S. Langerman, and B. Speckmann.
Computational Geometry, 47(2, Part C):296—306, 2014.

－ Abstract

We propose a new model of realistic input: k-guardable objects. An object is k-guardable if its boundary can be seen by k guards. We show that k-guardable polygons generalize two previously identified classes of realistic input. Following this, we give two simple algorithms for triangulating k-guardable polygons. One algorithm requires the guards as input while the other does not. Both take linear time assuming that k is constant and both are easily implementable.
－ BibTeX
```
@article{triangulating-and-guarding-realistic-polygons:2014,
    title = {Triangulating and Guarding Realistic Polygons},
    author = {G. Aloupis and P. Bose and V. Dujmovic and C.M. Gray and S. Langerman and B. Speckmann},
    year = {2014},
    bookTitle = {Computational Geometry},
}
```
[ PDF ]
Column Planarity and Partial Simultaneous Geometric Embedding
W.S. Evans, V.J.J. Kusters, M. Saumell, and B. Speckmann.
Graph Drawing (22nd International Symposium, GD 2014, Würzburg, Germany, September 24-26, 2014, Revised Selected Papers), pp. 259—271, 2014.

－ Abstract

We introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally, we say that R is column planar in G if we can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity. We prove near tight bounds for column planar subsets of trees: any tree on n vertices contains a column planar set of size at least 14n/17 and for any e¿>¿0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at most (5/6¿+¿e)n. We also consider a relaxation of simultaneous geometric embedding (SGE), which we call partial SGE (PSGE). A PSGE of two graphs G 1 and G 2 allows some of their vertices to map to two different points in the plane. We show how to use column planar subsets to construct k-PSGEs in which k vertices are still mapped to the same point. In particular, we show that any two trees on n vertices admit an 11n/17-PSGE, two outerpaths admit an n/4-PSGE, and an outerpath and a tree admit a 11n/34-PSGE.
－ BibTeX
```
@article{column-planarity-and-partial-simultaneous-geometric-embedding:2014,
    title = {Column Planarity and Partial Simultaneous Geometric Embedding},
    author = {W.S. Evans and V.J.J. Kusters and M. Saumell and B. Speckmann},
    year = {2014},
    bookTitle = {Graph Drawing (22nd International Symposium, GD 2014, Würzburg, Germany, September 24-26, 2014, Revised Selected Papers)},
}
```
[ PDF ]
Computing the Fréchet Distance with Shortcuts is NP-Hard
M. Buchin, A. Driemel, and B. Speckmann.
30th ACM Symposium on Computational Geometry (SoCG, Kyoto, Japan, June 8-11, 2014), pp. 367—376, 2014.

－ Abstract

We study the shortcut Fréchet distance, a natural variant of the Fréchet distance, that allows us to take shortcuts from and to any point along one of the curves. The classic Fréchet distance is a bottle-neck distance measure and hence quite sensitive to outliers. The shortcut Fréchet distance allows us to cut across outliers and hence produces more meaningful results when dealing with real world data. Driemel and Har-Peled recently described approximation algorithms for the restricted case where shortcuts have to start and end at input vertices. We show that, in the general case, the problem of computing the shortcut Fréchet distance is NP-hard. This is the first hardness result for a variant of the Fréchet distance between two polygonal curves in the plane. We also present two algorithms for the decision problem: a 3-approximation algorithm for the general case and an exact algorithm for the vertex-restricted case. Both algorithms run in O(n3 log n) time.
－ BibTeX
```
@article{computing-the-fr-chet-distance-with-shortcuts-is-np-hard:2014,
    title = {Computing the Fréchet Distance with Shortcuts is NP-Hard},
    author = {M. Buchin and A. Driemel and B. Speckmann},
    year = {2014},
    bookTitle = {30th ACM Symposium on Computational Geometry (SoCG, Kyoto, Japan, June 8-11, 2014)},
}
```
[ PDF ]
Exploring Curved Schematization
A.I. van Goethem, W. Meulemans, B. Speckmann, and J.D. Wood.
7th IEEE Pacific Visualization Symposium (PacificVis), pp. 1—8, 2014.

－ Abstract

Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization most previous work focuses on straight lines. We present a new algorithm for area-preserving curved schematization of geographic outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs. We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of 'curviness': we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. We conducted an online user study investigating the effectiveness of the curved schematizations compared to straight-line schematizations of equivalent complexity. While the visual complexity of the curved shapes was judged higher than those using straight lines, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents.
－ BibTeX
```
@article{exploring-curved-schematization:2014,
    title = {Exploring Curved Schematization},
    author = {A.I. van Goethem and W. Meulemans and B. Speckmann and J.D. Wood},
    year = {2014},
    bookTitle = {7th IEEE Pacific Visualization Symposium (PacificVis)},
}
```
[ PDF ]

Geometric Kth Shortest Paths: the Applet
J. Hershberger, V. Polishchuk, B. Speckmann, and T. Talvitie.
30th ACM Symposium on Computational Geometry (SoCG, Kyoto, Japan, June 8-11, 2014), pp. 96—97, 2014.

－ Abstract

－ BibTeX

@article{geometric-kth-shortest-paths-the-applet:2014,
    title = {Geometric Kth Shortest Paths: the Applet},
    author = {J. Hershberger and V. Polishchuk and B. Speckmann and T. Talvitie},
    year = {2014},
    bookTitle = {30th ACM Symposium on Computational Geometry (SoCG, Kyoto, Japan, June 8-11, 2014)},
}

GlamMap: Visualising Library Metadata
A. Betti, D.H.P. Gerrits, B. Speckmann, and H. Berg, van den.
VALA 2014 (17th Biennual Conference and Exhibition, Melbourne, Australia, February 3-6, 2014), pp. 1—15, 2014.

－ Abstract

Libraries provide access to large amounts of library metadata. Unfortunately, many libraries only offer textual interfaces for searching and browsing their holdings. Visualisations provide simpler, faster, and more efficient ways to navigate, search and study large quantities of metadata. This paper presents GlamMap, a visualisation tool that displays library metadata on an interactive, computer-generated geographic map. We provide detailed discussion of how GlamMap benefits the work of librarians and researchers. We show how geographic representations help librarians to perform tasks such as collection assessment and how geographic information helps researchers to identify important scientific resources.
－ BibTeX
```
@article{glammap-visualising-library-metadata:2014,
    title = {GlamMap: Visualising Library Metadata},
    author = {A. Betti and D.H.P. Gerrits and B. Speckmann and H. Berg, van den},
    year = {2014},
    bookTitle = {VALA 2014 (17th Biennual Conference and Exhibition, Melbourne, Australia, February 3-6, 2014)},
}
```
[ PDF ]
Map Schematization with Circular Arcs
T. van Dijk, A.I. van Goethem, J.H. Haunert, W. Meulemans, and B. Speckmann.
8th International Conference on Geographic Information Science (GIScience), pp. 1—17, 2014.

－ Abstract

We present an algorithm to compute schematic maps with circular arcs. Our algorithm iteratively replaces two consecutive arcs with a single arc to reduce the complexity of the output map and thus to increase its level of abstraction. Our main contribution is a method for replacing arcs that meet at high-degree vertices. This allows us to greatly reduce the output complexity, even for dense networks. We experimentally evaluate the effectiveness of our algorithm in three scenarios: territorial outlines, road networks, and metro maps. For the latter, we combine our approach with an algorithm to more evenly distribute stations. Our experiments show that our algorithm produces high-quality results for territorial outlines and metro maps. However, the lack of caricature (exaggeration of typical features) makes it less useful for road networks.
－ BibTeX
```
@article{map-schematization-with-circular-arcs:2014,
    title = {Map Schematization with Circular Arcs},
    author = {T. van Dijk and A.I. van Goethem and J.H. Haunert and W. Meulemans and B. Speckmann},
    year = {2014},
    bookTitle = {8th International Conference on Geographic Information Science (GIScience)},
}
```
[ PDF ]
Packing Plane Spanning Trees and Paths in Complete Geometric Graphs
O. Aichholzer, T. Hackl, M. Korman, M.J. van Kreveld, M. Löffler, A. Pilz, B. Speckmann, and E. Welzl.
26th Canadian Conference on Computational Geometry (CCCG 2014), 11-13 August 2014, Halifax, Canada, pp. 233—238, 2014.

－ Abstract

We consider the following question: How many edgedisjoint plane spanning trees are contained in a complete geometric graph GKn on any set S of n points in general position in the plane?
－ BibTeX
```
@article{packing-plane-spanning-trees-and-paths-in-complete-geometric-graphs:2014,
 title = {Packing Plane Spanning Trees and Paths in Complete Geometric Graphs},
 author = {O. Aichholzer and T. Hackl and M. Korman and M.J. van Kreveld and M. Löffler and A. Pilz and B. Speckmann and E. Welzl},
 year = {2014},
 bookTitle = {26th Canadian Conference on Computational Geometry (CCCG 2014), 11-13 August 2014, Halifax, Canada},
}
```
[ PDF ]

Trajectory Grouping Structure: the Video
K. Buchin, M. Buchin, M.J. Kreveld, van, B. Speckmann, and F. Staals.
30th Annual Symposium on Computational Geometry (SOCG'14, Kyoto, Japan, June 8-11, 2014), pp. 88—89, 2014.

－ Abstract

－ BibTeX

@article{trajectory-grouping-structure-the-video:2014,
    title = {Trajectory Grouping Structure: the Video},
    author = {K. Buchin and M. Buchin and M.J. Kreveld, van and B. Speckmann and F. Staals},
    year = {2014},
    bookTitle = {30th Annual Symposium on Computational Geometry (SOCG'14, Kyoto, Japan, June 8-11, 2014)},
}

[ PDF ]

Travel-Time Maps: Linear Cartograms with Fixed Vertex Locations
K. Buchin, A.I. van Goethem, M. Hoffmann, M.J. van Kreveld, and B. Speckmann.
Proc. of the 8th International Conference on Geographic Information Science (GIScience), pp. 18—33, 2014.

－ Abstract

Linear cartograms visualize travel times between locations, usually by deforming the underlying map such that Euclidean distance corresponds to travel time. We introduce an alternative model, where the map and the locations remain fixed, but edges are drawn as sinusoid curves. Now the travel time over a road corresponds to the length of the curve. Of course the curves might intersect if not placed carefully. We study the corresponding algorithmic problem and show that suitable placements can be computed efficiently. However, the problem of placing as many curves as possible in an ideal, centered position is NP-hard. We introduce three heuristics to optimize the number of centered curves and show how to create animated visualizations.
－ BibTeX
```
@article{travel-time-maps-linear-cartograms-with-fixed-vertex-locations:2014,
    title = {Travel-Time Maps: Linear Cartograms with Fixed Vertex Locations},
    author = {K. Buchin and A.I. van Goethem and M. Hoffmann and M.J. van Kreveld and B. Speckmann},
    year = {2014},
    bookTitle = {Proc. of the 8th International Conference on Geographic Information Science (GIScience)},
}
```
[ PDF ]
Distance-Sensitive Point Location Made Easy
B. Aronov, M.T. Berg, de, D. Eppstein, M.J.M. Roeloffzen, and B. Speckmann.
pp. 1—4, 2014.

－ Abstract

Let S be a planar polygonal subdivision with n edges contained in the unit square. We present a data structure for point location in S where queries with points far away from any region boundary are answered faster. More precisely, we show that point location queries can be answered in time O(1 + min(log \frac{1}{\Delta_p} , log n)), where {\Delta_p} is the Euclidean distance of the query point p to the boundary of the region containing p. Our solution consists of a depth-bounded quadtree and a general point location structure, both of which can be constructed in O(n log n)time. We also show how to extend the result to convex polyhedral subdivisions in three dimensions.
－ BibTeX
```
@article{distance-sensitive-point-location-made-easy:2014,
    title = {Distance-Sensitive Point Location Made Easy},
    author = {B. Aronov and M.T. Berg, de and D. Eppstein and M.J.M. Roeloffzen and B. Speckmann},
    year = {2014},
    bookTitle = {},
}
```
[ PDF ]
Sea Regions for Rectangular Cartograms
K. Buchin, R.G. Cano, P.J. Rezende, de, C.C. Souza, de, and B. Speckmann.
pp. 1—4, 2014.

－ Abstract

In a rectangular cartogram, each region of a map is represented by a rectangle whose area is proportional to some statistical data of interest. Current techniques for constructing rectangular cartograms partition a large rectangle (the map) into a set of smaller rectangles which correspond to land or sea regions. The position and size of sea rectangles determine the outline of land masses. Therefore, sea regions have a direct impact on the recognizability and, thus, on the visual quality of cartograms. In this paper, we describe the first algorithm for the automated creation of sea regions for rectangular cartograms and present results obtained with our method.
－ BibTeX
```
@article{sea-regions-for-rectangular-cartograms:2014,
    title = {Sea Regions for Rectangular Cartograms},
    author = {K. Buchin and R.G. Cano and P.J. Rezende, de and C.C. Souza, de and B. Speckmann},
    year = {2014},
    bookTitle = {},
}
```
[ PDF ]

2013

Flip Graphs of Bounded-Degree Triangulations
O. Aichholzer, T. Hackl, D. Orden, P. Ramos, G. Rote, A. Schulz, and B. Speckmann.
Graphs and Combinatorics, 29(6):1577—1593, 2013.

－ Abstract

We study ¿ip graphs of triangulations whose maximum vertex degree is bounded by a constant k. In particular, we consider triangulations of sets of n points in convex position in the plane and prove that their ¿ip graph is connected if and only if k > 6; the diameter of the ¿ip graph is O(n2). We also show that, for general point sets, ¿ip graphs of pointed pseudo-triangulations can be disconnected for k = 9, and ¿ip graphs of triangulations can be disconnected for any k. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound k by a small constant. Any two triangulations with maximum degree at most k of a convex point set are connected in the ¿ip graph by a path of length O(n log n), where every intermediate triangulation has maximum degree at most k + 4. Keywords: Flip graphs - Triangulations - Rotation distance - Connectivity - Degree bounds
－ BibTeX
```
@article{flip-graphs-of-bounded-degree-triangulations:2013,
    title = {Flip Graphs of Bounded-Degree Triangulations},
    author = {O. Aichholzer and T. Hackl and D. Orden and P. Ramos and G. Rote and A. Schulz and B. Speckmann},
    year = {2013},
    bookTitle = {Graphs and Combinatorics},
}
```
KelpFusion: a Hybrid Set Visualization Technique
W. Meulemans, N. Henry Riche, B. Speckmann, B. Alper, and T. Dwyer.
IEEE Transactions on Visualization and Computer Graphics, 19(11):1846—1858, 2013.

－ Abstract

We present KelpFusion: a method for depicting set membership of items on a map or other visualization using continuous boundaries. KelpFusion is a hybrid representation that bridges hull techniques such as Bubble Sets and Euler Diagrams and line- and graph-based techniques such as LineSets and Kelp Diagrams. We describe an algorithm based on shortest-path graphs to compute KelpFusion visualizations. Based on a single parameter, the shortest-path graph varies from the minimal spanning tree to the convex hull of a point set. Shortest-path graphs aim to capture the shape of a point set and smoothly adapt to sets of varying densities. KelpFusion fills enclosed faces based on a set of simple legibility rules. We present the results of a controlled experiment comparing KelpFusion to Bubble Sets and LineSets. We conclude that KelpFusion outperforms Bubble Sets both in accuracy and completion time, and outperforms LineSets in completion time. Keywords: Information visualization, visualization techniques and methodologies
－ BibTeX
```
@article{kelpfusion-a-hybrid-set-visualization-technique:2013,
    title = {KelpFusion: a Hybrid Set Visualization Technique},
    author = {W. Meulemans and N. Henry Riche and B. Speckmann and B. Alper and T. Dwyer},
    year = {2013},
    bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
}
```
[ PDF ]
Maximizing Maximal Angles for Plane Straight-Line Graphs
O. Aichholzer, T. Hackl, M. Hoffmann, C. Huemer, F. Santos, B. Speckmann, and B. Vogtenhuber.
Computational Geometry, 46(1):17—28, 2013.

－ Abstract

Let G=(S,E) be a plane straight-line graph on a finite point set S¿R2 in general position. The incident angles of a point p¿S in G are the angles between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight-line graph is called f-open if each vertex has an incident angle of size at least f. In this paper we study the following type of question: What is the maximum angle f such that for any finite set S¿R2 of points in general position we can find a graph from a certain class of graphs on S that is f-open? In particular, we consider the classes of triangulations, spanning trees, and spanning paths on S and give tight bounds in most cases.
－ BibTeX
```
@article{maximizing-maximal-angles-for-plane-straight-line-graphs:2013,
    title = {Maximizing Maximal Angles for Plane Straight-Line Graphs},
    author = {O. Aichholzer and T. Hackl and M. Hoffmann and C. Huemer and F. Santos and B. Speckmann and B. Vogtenhuber},
    year = {2013},
    bookTitle = {Computational Geometry},
}
```
Topologically Safe Curved Schematisation
A.I. Goethem, van, W. Meulemans, A. Reimer, H.J. Haverkort, and B. Speckmann.
The Cartographic Journal, 50(3):276—285, 2013.

－ Abstract

Traditionally schematised maps make extensive use of curves. However, automated methods for schematisation are mostly restricted to straight lines. We present a generic framework for topology-preserving curved schematisation that allows a choice of quality measures and curve types. The framework fits a curve to every part of the input. It uses Voronoi diagrams to ensure that curves fitted to disjoint parts do not intersect. The framework then employs a dynamic program to find an optimal schematisation using the fitted curves. Our fully-automated approach does not need critical points or salient features. We illustrate our framework with Bézier curves and circular arcs.
－ BibTeX
```
@article{topologically-safe-curved-schematisation:2013,
    title = {Topologically Safe Curved Schematisation},
    author = {A.I. Goethem, van and W. Meulemans and A. Reimer and H.J. Haverkort and B. Speckmann},
    year = {2013},
    bookTitle = {The Cartographic Journal},
}
```
[ PDF ]
Accentuating Focus Maps Via Partial Schematization
T. van Dijk, A.I. van Goethem, J.H. Haunert, W. Meulemans, and B. Speckmann.
21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM GIS), pp. 418—421, 2013.

－ Abstract

We present an algorithm for schematized focus maps. Focus maps integrate a high detailed, enlarged focus region continuously in a given base map. Recent methods integrate both with such low distortion that the focus region becomes hard to identify. We combine focus maps with partial schematization to display distortion of the context and to emphasize the focus region. Schematization visually conveys geographical accuracy, while not increasing map complexity. We extend the focus-map algorithm to incorporate geometric proximity relationships and show how to combine focus maps with schematization in order to cater to different use cases.
－ BibTeX
```
@article{accentuating-focus-maps-via-partial-schematization:2013,
    title = {Accentuating Focus Maps Via Partial Schematization},
    author = {T. van Dijk and A.I. van Goethem and J.H. Haunert and W. Meulemans and B. Speckmann},
    year = {2013},
    bookTitle = {21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM GIS)},
}
```
[ PDF ]
Colored Spanning Graphs for Set Visualization
F. Hurtado, M. Korman, M.J. Kreveld, van, M. Löffler, V. Sacristán, R.I. Silveira, and B. Speckmann.
Graph Drawing : 21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013, Revised Selected Papers, pp. 280—291, 2013.

－ Abstract

We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A red-blue-purple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem is NP-hard. Hence we give an (½ ¿ + 1)-approximation, where ¿ is the Steiner ratio. We also present efficient exact solutions if the points are located on a line or a circle. Finally we consider extensions to more than two sets.
－ BibTeX
```
@article{colored-spanning-graphs-for-set-visualization:2013,
    title = {Colored Spanning Graphs for Set Visualization},
    author = {F. Hurtado and M. Korman and M.J. Kreveld, van and M. Löffler and V. Sacristán and R.I. Silveira and B. Speckmann},
    year = {2013},
    bookTitle = {Graph Drawing : 21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013, Revised Selected Papers},
}
```
Distance-Sensitive Planar Point Location
B. Aronov, M.T. Berg, de, M.J.M. Roeloffzen, and B. Speckmann.
Algorithms and Data Structures (13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings), pp. 49—60, 2013.

－ Abstract

Let be a connected planar polygonal subdivision with n edges and of total area 1. We present a data structure for point location in TeX where queries with points far away from any region boundary are answered faster. More precisely, we show that point location queries can be answered in time TeX, where ¿p is the distance of the query point p to the boundary of the region containing p. Our structure is based on the following result: any simple polygon P can be decomposed into a linear number of convex quadrilaterals with the following property: for any point p¿¿¿P, the quadrilateral containing p has area TeX.
－ BibTeX
```
@article{distance-sensitive-planar-point-location:2013,
    title = {Distance-Sensitive Planar Point Location},
    author = {B. Aronov and M.T. Berg, de and M.J.M. Roeloffzen and B. Speckmann},
    year = {2013},
    bookTitle = {Algorithms and Data Structures (13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings)},
}
```
Improved Grid Map Layout by Point Set Matching
D. Eppstein, M.J. Kreveld, van, B. Speckmann, and F. Staals.
6th IEEE Pacific Visualization Symposium (PacificVis, Sydney, Australia, February 26-March 1, 2013), pp. 25—32, 2013.

－ Abstract

Associating the regions of a geographic subdivision with the cells of a grid is a basic operation that is used in various types of maps, like spatially ordered treemaps and OD maps. In these cases the regular shapes of the grid cells allows easy representation of extra information about the regions. The main challenge is to find an association that allows a user to find a region in the grid quickly. We call the representation of a set of regions as a grid a grid map.
－ BibTeX
```
@article{improved-grid-map-layout-by-point-set-matching:2013,
    title = {Improved Grid Map Layout by Point Set Matching},
    author = {D. Eppstein and M.J. Kreveld, van and B. Speckmann and F. Staals},
    year = {2013},
    bookTitle = {6th IEEE Pacific Visualization Symposium (PacificVis, Sydney, Australia, February 26-March 1, 2013)},
}
```
Kinetic 2-Centers in the Black-Box Model
M.T. Berg, de, M.J.M. Roeloffzen, and B. Speckmann.
Proc. 29th ACM Symposium on Computational Geometry (SoCG), pp. 145—154, 2013.

－ Abstract

We study two versions of the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P; the rectilinear 2-center problem correspondingly asks for two congruent axis-aligned squares of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_{max} on the maximum displacement of any point within one time step. We show how to maintain the rectilinear 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. For the Euclidean 2-center we give a similar result: we can maintain in amortized sub-linear time (again under certain assumptions on the distribution) a (1+e)-approximation of the optimal 2-center. In many cases---namely when the distance between the centers of the disks is relatively large or relatively small---the solution we maintain is actually optimal. We also present results for the case where the maximum speed of the centers is bounded. We describe a simple scheme to maintain a 2-approximation of the rectilinear 2-center, and we provide a scheme which gives a better approximation factor depending on several parameters of the point set and the maximum allowed displacement of the centers. The latter result can be used to obtain a 2.29-approximation for the Euclidean 2-center; this is an improvement over the previously best known bound of 8/p approx 2.55. These algorithms run in amortized sub-linear time per time step, as before under certain assumptions on the distribution.
－ BibTeX
```
@article{kinetic-2-centers-in-the-black-box-model:2013,
    title = {Kinetic 2-Centers in the Black-Box Model},
    author = {M.T. Berg, de and M.J.M. Roeloffzen and B. Speckmann},
    year = {2013},
    bookTitle = {Proc. 29th ACM Symposium on Computational Geometry (SoCG)},
}
```
Strict Confluent Drawing
D. Eppstein, D.H.R. Holten, M. Löffler, M. Nöllenburg, B. Speckmann, and K.A.B. Verbeek.
Graph Drawing (21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013. Revised Selected Papers), pp. 352—363, 2013.

－ Abstract

We define strict confluent drawing, a form of confluent drawing in which the existence of an edge is indicated by the presence of a smooth path through a system of arcs and junctions (without crossings), and in which such a path, if it exists, must be unique. We prove that it is NP-complete to determine whether a given graph has a strict confluent drawing but polynomial to determine whether it has an outerplanar strict confluent drawing with a fixed vertex ordering (a drawing within a disk, with the vertices placed in a given order on the boundary).
－ BibTeX
```
@article{strict-confluent-drawing:2013,
    title = {Strict Confluent Drawing},
    author = {D. Eppstein and D.H.R. Holten and M. Löffler and M. Nöllenburg and B. Speckmann and K.A.B. Verbeek},
    year = {2013},
    bookTitle = {Graph Drawing (21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013. Revised Selected Papers)},
}
```
Trajectory Grouping Structure
K. Buchin, M. Buchin, M.J. Kreveld, van, B. Speckmann, and F. Staals.
Algorithms and Data Structures (13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings), pp. 219—230, 2013.

－ Abstract

The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept from topology. The trajectory grouping structure has three natural parameters, namely group size, group duration, and entity inter-distance. These parameters allow us to obtain detailed or global views of the data. We prove complexity bounds on the maximum number of maximal groups that can be present, and give algorithms to compute the grouping structure efficiently. Furthermore, we showcase the results of experiments using data generated by the NetLogo flocking model and from the Starkey project. Although there is no ground truth for the groups in this data, the experiments show that the trajectory grouping structure is plausible and has the desired effects when changing the essential parameters. Our research provides the first complete study of trajectory group evolvement, including combinatorial, algorithmic, and experimental results.
－ BibTeX
```
@article{trajectory-grouping-structure:2013,
    title = {Trajectory Grouping Structure},
    author = {K. Buchin and M. Buchin and M.J. Kreveld, van and B. Speckmann and F. Staals},
    year = {2013},
    bookTitle = {Algorithms and Data Structures (13th International Symposium, WADS 2013, London, ON, Canada, August 12-14, 2013. Proceedings)},
}
```
Computing the Fréchet Distance with Shortcuts is NP-Hard
M. Buchin, A. Driemel, and B. Speckmann.
pp. 43—46, 2013.

－ Abstract

We study the shortcut Fréchet distance, a natural variant of the Fréchet distance that allows us to take shortcuts from and to any point along one of the curves. We show that, surprisingly, the problem of computing the shortcut Fréchet distance exactly is NP-hard. Furthermore, we give a 3-approximation algorithm for the decision version of the problem.
－ BibTeX
```
@article{computing-the-fr-chet-distance-with-shortcuts-is-np-hard:2013,
    title = {Computing the Fréchet Distance with Shortcuts is NP-Hard},
    author = {M. Buchin and A. Driemel and B. Speckmann},
    year = {2013},
    bookTitle = {},
}
```
[ PDF ]
Kinetic Euclidean 2-Centers in the Black-Box Model
M.T. Berg, de, M.J.M. Roeloffzen, and B. Speckmann.
pp. 173—176, 2013.

－ Abstract

We study the 2-center problem for moving points in the plane. Given a set P of n points, the Euclidean 2-center problem asks for two congruent disks of minimum size that together cover P. Our methods work in the black-box KDS model, where we receive the locations of the points at regular time steps and we know an upper bound d_max on the maximum displacement of any point within one time step. We show how to maintain a (1 + e)-approximation of the Euclidean 2-center in amortized sub-linear time per time step, under certain assumptions on the distribution of the point set P. In many cases --namely when the distance between the centers of the disks is relatively large or relatively small-- the solution we maintain is actually optimal.
－ BibTeX
```
@article{kinetic-euclidean-2-centers-in-the-black-box-model:2013,
    title = {Kinetic Euclidean 2-Centers in the Black-Box Model},
    author = {M.T. Berg, de and M.J.M. Roeloffzen and B. Speckmann},
    year = {2013},
    bookTitle = {},
}
```
[ PDF ]
Topologically Safe Curved Schematization
A.I. van Goethem, H.J. Haverkort, W. Meulemans, A. Reimer, and B. Speckmann.
pp. 87—90, 2013.

－ Abstract

Traditionally schematized maps make extensive use of curves. However, automated methods for schematization are mostly restricted to straight lines. We present a generic framework for topology-preserving curved schematization that allows a choice of quality measures and curve types. Our fully-automated approach does not need critical points or salient features. We illustrate our framework with Bézier curves and circular arcs.
－ BibTeX
```
@article{topologically-safe-curved-schematization:2013,
    title = {Topologically Safe Curved Schematization},
    author = {A.I. van Goethem and H.J. Haverkort and W. Meulemans and A. Reimer and B. Speckmann},
    year = {2013},
    bookTitle = {},
}
```
[ PDF ]
Strict Confluent Drawing
D. Eppstein, D.H.R. Holten, M. Löffler, M. Nöllenburg, B. Speckmann, and K.A.B. Verbeek.
2013.

－ Abstract

We define strict confluent drawing, a form of confluent drawing in which the existence of an edge is indicated by the presence of a smooth path through a system of arcs and junctions (without crossings), and in which such a path, if it exists, must be unique. We prove that it is NP-complete to determine whether a given graph has a strict confluent drawing but polynomial to determine whether it has an outerplanar strict confluent drawing with a fixed vertex ordering (a drawing within a disk, with the vertices placed in a given order on the boundary).
－ BibTeX
```
@article{strict-confluent-drawing:2013,
    title = {Strict Confluent Drawing},
    author = {D. Eppstein and D.H.R. Holten and M. Löffler and M. Nöllenburg and B. Speckmann and K.A.B. Verbeek},
    year = {2013},
    bookTitle = {},
}
```
[ PDF ]
Trajectory Grouping Structure
K. Buchin, M. Buchin, M.J. Kreveld, van, B. Speckmann, and F. Staals.
2013.

－ Abstract

The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept from topology. The trajectory grouping structure has three natural parameters that allow more global views of the data in group size, group duration, and entity inter-distance. We prove complexity bounds on the maximum number of maximal groups that can be present, and give algorithms to compute the grouping structure efficiently. We also study how the trajectory grouping structure can be made robust, that is, how brief interruptions of groups can be disregarded in the global structure, adding a notion of persistence to the structure. Furthermore, we showcase the results of experiments using data generated by the NetLogo flocking model and from the Starkey project. The Starkey data describe the movement of elk, deer, and cattle. Although there is no ground truth for the grouping structure in this data, the experiments show that the trajectory grouping structure is plausible and has the desired effects when changing the essential parameters. Our research provides the first complete study of trajectory group evolvement, including combinatorial, algorithmic, and experimental results.
－ BibTeX
```
@article{trajectory-grouping-structure:2013,
    title = {Trajectory Grouping Structure},
    author = {K. Buchin and M. Buchin and M.J. Kreveld, van and B. Speckmann and F. Staals},
    year = {2013},
    bookTitle = {},
}
```
[ PDF ]

Convex-Arc Drawings of Pseudolines
D. Eppstein, M. Garderen, van, B. Speckmann, and T. Ueckerdt.
pp. 522—523, 2013.

－ BibTeX

@article{convex-arc-drawings-of-pseudolines:2013,
    title = {Convex-Arc Drawings of Pseudolines},
    author = {D. Eppstein and M. Garderen, van and B. Speckmann and T. Ueckerdt},
    year = {2013},
    bookTitle = {},
}

2012

Analysis and Visualization of Animal Movement
J. Shamoun-Baranes, E.E. Loon, van, R.S. Purves, B. Speckmann, D. Weiskopf, and C.J. Camphuysen.
Biology Letters, 8(1):6—9, 2012.

－ Abstract

The interdisciplinary workshop ‘Analysis and Visualization of Moving Objects’ was held at the Lorentz Centre in Leiden, The Netherlands, from 27 June to 1 July 2011. It brought together international specialists from ecology, computer science and geographical information science actively involved in the exploration, visualization and analysis of moving objects, such as marine reptiles, mammals, birds, storms, ships, cars and pedestrians. The aim was to share expertise, methodologies, data and common questions between different fields, and to work towards making significant advances in movement research. A data challenge based on GPS tracking of lesser black-backed gulls (Larus fuscus) was used to stimulate initial discussions, cross-fertilization between research groups and to serve as an initial focus for activities during the workshop.
－ BibTeX
```
@article{analysis-and-visualization-of-animal-movement:2012,
    title = {Analysis and Visualization of Animal Movement},
    author = {J. Shamoun-Baranes and E.E. Loon, van and R.S. Purves and B. Speckmann and D. Weiskopf and C.J. Camphuysen},
    year = {2012},
    bookTitle = {Biology Letters},
}
```
Area-Universal and Constrained Rectangular Layouts
D. Eppstein, E. Mumford, B. Speckmann, and K.A.B. Verbeek.
SIAM Journal on Computing, 41(3):537—564, 2012.

－ Abstract

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. These layouts are used as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it is desirable for one rectangular layout to represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists. Furthermore we show how to impose restrictions on the orientations of edges and junctions of the rectangular layout. Such an orientation-constrained layout, if it exists, may be constructed in polynomial time, and all orientation-constrained layouts may be listed in polynomial time per layout.
－ BibTeX
```
@article{area-universal-and-constrained-rectangular-layouts:2012,
    title = {Area-Universal and Constrained Rectangular Layouts},
    author = {D. Eppstein and E. Mumford and B. Speckmann and K.A.B. Verbeek},
    year = {2012},
    bookTitle = {SIAM Journal on Computing},
}
```
[ PDF ]
Kelp Diagrams: Point Set Membership Visualization
K. Dinkla, M.J. Kreveld, van, B. Speckmann, and M.A. Westenberg.
Computer Graphics Forum, 31(3):875—884, 2012.

－ Abstract

We present Kelp Diagrams, a novel method to depict set relations over points, i.e., elements with predefined positions. Our method creates schematic drawings and has been designed to take aesthetic quality, efficiency, and effectiveness into account. This is achieved by a routing algorithm, which links elements that are part of the same set by constructing minimum cost paths over a tangent visibility graph. There are two styles of Kelp Diagrams to depict overlapping sets, a nested and a striped style, each with its own strengths and weaknesses. We compare Kelp Diagrams with two existing methods and show that our approach provides a more consistent and clear depiction of both element locations and their set relations.
－ BibTeX
```
@article{kelp-diagrams-point-set-membership-visualization:2012,
    title = {Kelp Diagrams: Point Set Membership Visualization},
    author = {K. Dinkla and M.J. Kreveld, van and B. Speckmann and M.A. Westenberg},
    year = {2012},
    bookTitle = {Computer Graphics Forum},
}
```
Kinetic Convex Hulls, Delaunay Triangulations and Connectivity Structures in the Black-Box Model
M.T. Berg, de, M.J.M. Roeloffzen, and B. Speckmann.
Journal of Computational Geometry, 3(1):222—249, 2012.

－ Abstract

Over the past decade, the kinetic-data-structures framework has become thestandard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum displacement of any point in one time step. We study the maintenance of the convex hull and the Delaunay triangulation of a planar point set P in the black-box model, under the following assumption on dmax: there is some constant k such that for any point p in P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ¿k, the so-called k-spread of P. We show how to update the convex hull at each time step in O(min(n, k¿klog n)log n) amortized time. For the Delaunay triangulation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k2¿k2) at each time step.
－ BibTeX
```
@article{kinetic-convex-hulls-delaunay-triangulations-and-connectivity-structures-in-the-black-box-model:2012,
    title = {Kinetic Convex Hulls, Delaunay Triangulations and Connectivity Structures in the Black-Box Model},
    author = {M.T. Berg, de and M.J.M. Roeloffzen and B. Speckmann},
    year = {2012},
    bookTitle = {Journal of Computational Geometry},
}
```
[ PDF ]
Shooting Permanent Rays Among Disjoint Polygons in the Plane
M. Ishaque, B. Speckmann, and Cs.D. Tóth.
SIAM Journal on Computing, 41(4):1005—1027, 2012.

－ Abstract

We present a data structure for ray shooting and insertion in the free space between disjoint polygonal obstacles with a total of $n$ vertices in the plane, where each ray starts at the boundary of some obstacle. The portion of each query ray between the starting point and the first obstacle hit is inserted permanently as a new obstacle. Our data structure uses $O(n\log n)$ space and preprocessing time, and it supports $m$ successive ray shooting and insertion queries in $O((n+m)\log^2 n + m\log m)$ total time in the real RAM model of computation. We present two applications: (1) Our data structure supports efficient implementation of auto-partitions in the plane, that is, binary space partitions where each partition is done along the supporting line of an input segment. If $n$ input line segments are fragmented into $m$ pieces by an auto-partition, then it can now be implemented in $O(n\log^2n+m\log m)$ time. This improves the expected runtime of Patersen and Yao's classical randomized auto-partition algorithm for $n$ disjoint line segments in the plane to $O(n\log^2 n)$. (2) If we are given disjoint polygonal obstacles with a total of $n$ vertices in the plane, a permutation of the reflex vertices, and a half-line at each reflex vertex that partitions the reflex angle into two convex angles, then the convex partitioning algorithm draws a ray emanating from each reflex vertex in the prescribed order in the given direction until it hits another obstacle, a previous ray, or infinity. The previously best implementation (with a semidynamic ray shooting data structure) requires $O(n^{3/2-\varepsilon/2})$ time using $O(n^{1+\varepsilon})$ space for any $\varepsilon>0$. Our data structure improves the runtime to $O(n\log^2 n)$.
－ BibTeX
```
@article{shooting-permanent-rays-among-disjoint-polygons-in-the-plane:2012,
    title = {Shooting Permanent Rays Among Disjoint Polygons in the Plane},
    author = {M. Ishaque and B. Speckmann and Cs.D. Tóth},
    year = {2012},
    bookTitle = {SIAM Journal on Computing},
}
```
[ PDF ]
Context-Aware Similarity of Trajectories
M. Buchin, S. Dodge, and B. Speckmann.
Geographic Information Science (7th International Conference, GIScience 2012, Columbus, OH, USA, September 18-21, 2012. Proceedings), pp. 43—56, 2012.

－ Abstract

The movement of animals, people, and vehicles is embedded in a geographic context. This context influences the movement. Most analysis algorithms for trajectories have so far ignored context, which severely limits their applicability. In this paper we present a model for geographic context that allows us to integrate context into the analysis of movement data. Based on this model we develop simple but efficient context-aware similarity measures. We validate our approach by applying these measures to hurricane trajectories. Keywords: Movement data – geographic context – similarity measures
－ BibTeX
```
@article{context-aware-similarity-of-trajectories:2012,
    title = {Context-Aware Similarity of Trajectories},
    author = {M. Buchin and S. Dodge and B. Speckmann},
    year = {2012},
    bookTitle = {Geographic Information Science (7th International Conference, GIScience 2012, Columbus, OH, USA, September 18-21, 2012. Proceedings)},
}
```
Evolution Strategies for Optimizing Rectangular Cartograms
K. Buchin, B. Speckmann, and S. Verdonschot.
Geographic Information Science (7th International Conference, GIScience 2012, Columbus, OH, USA, September 18-21, 2012. Proceedings), pp. 29—42, 2012.

－ Abstract

A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable such as population or GDP. In recent years several algorithms for the automated construction of rectangular cartograms have been proposed, some of which are based on rectangular duals of the dual graph of the input map. In this paper we present a new approach to efficiently search within the exponentially large space of all possible rectangular duals. We employ evolution strategies that find rectangular duals which can be used for rectangular cartograms with correct adjacencies and (close to) zero cartographic error. This is a considerable improvement upon previous methods that have to either relax adjacency requirements or deal with larger errors. We present extensive experimental results for a large variety of data sets. Keywords: Rectangular cartogram – evolution strategy – regular edge labeling
－ BibTeX
```
@article{evolution-strategies-for-optimizing-rectangular-cartograms:2012,
    title = {Evolution Strategies for Optimizing Rectangular Cartograms},
    author = {K. Buchin and B. Speckmann and S. Verdonschot},
    year = {2012},
    bookTitle = {Geographic Information Science (7th International Conference, GIScience 2012, Columbus, OH, USA, September 18-21, 2012. Proceedings)},
}
```
Kinetic Compressed Quadtrees in the Black-Box Model with Applications to Collision Detection for Low-Density Scenes
M.T. Berg, de, M.J.M. Roeloffzen, and B. Speckmann.
Algorithms - ESA 2012 (20th European Symposium on Algorithms, Ljubljana, Slovenia, September 10-12, 2012. Proceedings), pp. 383—394, 2012.

－ Abstract

We present an efficient method for maintaining a compressed quadtree for a set of moving points in R d . Our method works in the black-box KDS model, where we receive the locations of the points at regular time steps and we know a bound d max on the maximum displacement of any point within one time step. When the number of points within any ball of radius d max is at most k at any time, then our update algorithm runs in O(nlogk) time. We generalize this result to constant-complexity moving objects in R d . The compressed quadtree we maintain has size O(n); under similar conditions as for the case of moving points it can be maintained in O(n log¿) time per time step, where ¿ is the density of the set of objects. The compressed quadtree can be used to perform broad-phase collision detection for moving objects; it will report in O((¿¿+¿k)n) time a superset of all intersecting pairs of objects.
－ BibTeX
```
@article{kinetic-compressed-quadtrees-in-the-black-box-model-with-applications-to-collision-detection-for-low-density-scenes:2012,
    title = {Kinetic Compressed Quadtrees in the Black-Box Model with Applications to Collision Detection for Low-Density Scenes},
    author = {M.T. Berg, de and M.J.M. Roeloffzen and B. Speckmann},
    year = {2012},
    bookTitle = {Algorithms - ESA 2012 (20th European Symposium on Algorithms, Ljubljana, Slovenia, September 10-12, 2012. Proceedings)},
}
```
Locally Correct Fréchet Matchings
K. Buchin, M. Buchin, W. Meulemans, and B. Speckmann.
20th European Symposium on Algorithms (ESA), pp. 229—240, 2012.

－ Abstract

The Fréchet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Fréchet distance a Fréchet matching. There are often many different Fréchet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Fréchet matchings to "natural" matchings and to this end introduce locally correct Fréchet matchings. We prove that at least one such matching exists for two polygonal curves and give an O(N^3 log N) algorithm to compute it, where N is the total number of edges in both curves. We also present an O(N^2) algorithm to compute a locally correct discrete Fréchet matching.
－ BibTeX
```
@article{locally-correct-fr-chet-matchings:2012,
    title = {Locally Correct Fréchet Matchings},
    author = {K. Buchin and M. Buchin and W. Meulemans and B. Speckmann},
    year = {2012},
    bookTitle = {20th European Symposium on Algorithms (ESA)},
}
```
[ PDF ]
Kinetic Collision Detection for Low-Density Scenes in the Black-Box Model
M.T. Berg, de, M.J.M. Roeloffzen, and B. Speckmann.
pp. 53—56, 2012.

－ Abstract

We present an efficient method for collision detection in the black-box KDS model for a set S of n objects in the plane. In this model we receive the object locations at regular time steps and we know a bound dmax on the maximum displacement of any object within one time step. Our method maintains, in O((¿+k)n) time per time step, a compressed quadtree on the bounding-box vertices of the objects; here ¿ denotes the density of S and k denotes the maximum number of objects that can intersect any disk of radius dmax. Collisions can then be detected by testing O((¿+k)2n) pairs of objects for intersection.
－ BibTeX
```
@article{kinetic-collision-detection-for-low-density-scenes-in-the-black-box-model:2012,
    title = {Kinetic Collision Detection for Low-Density Scenes in the Black-Box Model},
    author = {M.T. Berg, de and M.J.M. Roeloffzen and B. Speckmann},
    year = {2012},
    bookTitle = {},
}
```
[ PDF ]
Locally Correct Fréchet Matchings
K. Buchin, M. Buchin, W. Meulemans, and B. Speckmann.
pp. 81—84, 2012.

－ Abstract

The Fréchet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Fréchet distance a Fréchet matching. There are often many different Fréchet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Fréchet matchings to "natural" matchings and to this end introduce locally correct Fréchet matchings. We prove that at least one such a matching exists for two polygonal curves and give an algorithm to compute it.
－ BibTeX
```
@article{locally-correct-fr-chet-matchings:2012,
    title = {Locally Correct Fréchet Matchings},
    author = {K. Buchin and M. Buchin and W. Meulemans and B. Speckmann},
    year = {2012},
    bookTitle = {},
}
```
[ PDF ]
One-To-One Point Set Matchings for Grid Map Layout
D. Eppstein, M.J. van Kreveld, B. Speckmann, and F. Staals.
pp. 205—208, 2012.

－ Abstract

We study several one-to-one point set matching problems which are motivated by layout problems for grid maps. We are given two sets A and B of n points in the plane, and we wish to compute an optimal one-to-one matching between A and B. We consider two optimisation criteria: minimising the sum of the L1-distances between matched points, and maximising the number of pairs of points in A for which the matching preserves the directional relation. We show how to minimise the total L1-distance under translation or scaling in O(n6 log3 n) time, and under both translation and scaling in O(n10 log3 n) time. We further give a 4-approximation for preserving directional relations by computing a minimum L1-distance matching in O(n2 log3 n) time.
－ BibTeX
```
@article{one-to-one-point-set-matchings-for-grid-map-layout:2012,
    title = {One-To-One Point Set Matchings for Grid Map Layout},
    author = {D. Eppstein and M.J. van Kreveld and B. Speckmann and F. Staals},
    year = {2012},
    bookTitle = {},
}
```
[ PDF ]
Locally Correct Fréchet Matchings
K. Buchin, M. Buchin, W. Meulemans, and B. Speckmann.
2012.

－ Abstract

The Frechet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Frechet distance a Frechet matching. There are often many different Frechet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Frechet matchings to "natural" matchings and to this end introduce locally correct Frechet matchings. We prove that at least one such matching exists for two polygonal curves and give an O(N^3 log N) algorithm to compute it, where N is the total number of edges in both curves. We also present an O(N^2) algorithm to compute a locally correct discrete Frechet matching.
－ BibTeX
```
@article{locally-correct-fr-chet-matchings:2012,
    title = {Locally Correct Fréchet Matchings},
    author = {K. Buchin and M. Buchin and W. Meulemans and B. Speckmann},
    year = {2012},
    bookTitle = {},
}
```
[ PDF ]
Guest Editor's Foreword (Special Issue with Selected Papers from the 19th International Symposium on Graph Drawing, GD 2011)
M.J. Kreveld, van and B. Speckmann.
Journal of Graph Algorithms and Applications, 16(3):631—633, 2012.

－ Abstract

This issue of the Journal of Graph Algorithms and Applications is devoted to the nineteenth International Symposium on Graph Drawing, held September 19-21, 2011, in Eindhoven, the Netherlands.
－ BibTeX
```
@article{guest-editor-s-foreword-special-issue-with-selected-papers-from-the-19th-international-symposium-on-graph-drawing-gd-2011-:2012,
    title = {Guest Editor's Foreword (Special Issue with Selected Papers from the 19th International Symposium on Graph Drawing, GD 2011)},
    author = {M.J. Kreveld, van and B. Speckmann},
    year = {2012},
    bookTitle = {Journal of Graph Algorithms and Applications},
}
```
[ PDF ]

2011

Connect the Dot: Computing Feed-Links for Network Extension
B. Aronov, K. Buchin, M. Buchin, B.M.P. Jansen, T. Jong, de, M.J. Kreveld, van, M. Löffler, J. Luo, R.I. Silveira, and B. Speckmann.
Journal of Spatial Information Science, 3:3—31, 2011.

－ Abstract

Road network analysis can require distance from points that are not on the network themselves. We study the algorithmic problem of connecting a point inside a face (region) of the road network to its boundary while minimizing the detour factor of that point to any point on the boundary of the face. We show that the optimal single connection (feed-link) can be computed in O(¿7(n) log n) time, where n is the number of vertices that bounds the face and ¿7(n) is the slightly superlinear maximum length of a Davenport-Schinzel sequence of order 7. We also present approximation results for placing more feed-links, deal with the case that there are obstacles in the face of the road network that contains the point to be connected, and present various related results.
－ BibTeX
```
@article{connect-the-dot-computing-feed-links-for-network-extension:2011,
    title = {Connect the Dot: Computing Feed-Links for Network Extension},
    author = {B. Aronov and K. Buchin and M. Buchin and B.M.P. Jansen and T. Jong, de and M.J. Kreveld, van and M. Löffler and J. Luo and R.I. Silveira and B. Speckmann},
    year = {2011},
    bookTitle = {Journal of Spatial Information Science},
}
```
[ PDF ]
Empty Pseudo-Triangles in Point Sets
H.K. Ahn, S.W. Bae, M.J. Kreveld, van, I. Reinbacher, and B. Speckmann.
Discrete Applied Mathematics, 159(18):2205—2213, 2011.

－ Abstract

We study empty pseudo-triangles in a set P of n points in the plane, where an empty pseudo-triangle has its vertices at the points of P, and no points of P lie inside. We give bounds on the minimum and maximum number of empty pseudo-triangles. If P lies inside a triangle whose corners must be the convex vertices of the pseudo-triangle, then there can be between T(n2) and T(n3) empty pseudo-triangles. If the convex vertices of the pseudo-triangle are also chosen from P, this number lies between T(n3) and T(n6). If we count only star-shaped pseudo-triangles, the bounds are T(n2) and T(n5). We also study optimization problems: minimizing or maximizing the perimeter or the area over all empty pseudo-triangles defined by P. If P lies inside a triangle whose corners must be used, we can solve these problems in O(n3) time. In the general case, the running times are O(n6) for the maximization problems and O(nlogn) for the minimization problems.
－ BibTeX
```
@article{empty-pseudo-triangles-in-point-sets:2011,
    title = {Empty Pseudo-Triangles in Point Sets},
    author = {H.K. Ahn and S.W. Bae and M.J. Kreveld, van and I. Reinbacher and B. Speckmann},
    year = {2011},
    bookTitle = {Discrete Applied Mathematics},
}
```
Flow Map Layout Via Spiral Trees
K.A.B. Verbeek, K. Buchin, and B. Speckmann.
IEEE Transactions on Visualization and Computer Graphics, 17(12):2536—2544, 2011.

－ Abstract

Flow maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of flow between a source and a target. Good flow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding self-intersections. Most flow maps are still drawn by hand and only few automated methods exist. Some of the known algorithms do not support edgebundling and those that do, cannot guarantee crossing-free flows. We present a new algorithmic method that uses edge-bundling and computes crossing-free flows of high visual quality. Our method is based on so-called spiral trees, a novel type of Steiner tree which uses logarithmic spirals. Spiral trees naturally induce a clustering on the targets and smoothly bundle lines. Our flows can also avoid obstacles, such as map features, region outlines, or even the targets. We demonstrate our approach with extensive experiments.
－ BibTeX
```
@article{flow-map-layout-via-spiral-trees:2011,
    title = {Flow Map Layout Via Spiral Trees},
    author = {K.A.B. Verbeek and K. Buchin and B. Speckmann},
    year = {2011},
    bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
}
```
[ PDF ]
Geometric Simultaneous Embeddings of a Graph and a Matching
S. Cabello, M.J. Kreveld, van, G. Liotta, H. Meijer, B. Speckmann, and K.A.B. Verbeek.
Journal of Graph Algorithms and Applications, 15(1):79—96, 2011.

－ Abstract

The geometric simultaneous embedding problem asks whether two planar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous embedding is a current topic in graph drawing and positive and negative results are known for various classes of graphs. So far only connected graphs have been considered. In this paper we present the first results for the setting where one of the graphs is a matching. In particular, we show that there exist a planar graph and a matching which do not admit a geometric simultaneous embedding. This strengthens an analogous negative result for a planar graph and a path. On the positive side, we describe algorithms that compute a geometric simultaneous embedding of a matching and a wheel, outerpath, or tree. Our drawing algorithms minimize the number of orientations used to draw the edges of the matching. Specifically, when embedding a matching and a tree, we can draw all matching edges horizontally. When embedding a matching and a wheel or an outerpath, we use only two orientations.
－ BibTeX
```
@article{geometric-simultaneous-embeddings-of-a-graph-and-a-matching:2011,
    title = {Geometric Simultaneous Embeddings of a Graph and a Matching},
    author = {S. Cabello and M.J. Kreveld, van and G. Liotta and H. Meijer and B. Speckmann and K.A.B. Verbeek},
    year = {2011},
    bookTitle = {Journal of Graph Algorithms and Applications},
}
```
[ PDF ]
On Planar Supports for Hypergraphs
K. Buchin, M.J. Kreveld, van, H. Meijer, B. Speckmann, and K.A.B. Verbeek.
Journal of Graph Algorithms and Applications, 15(4):533—549, 2011.

－ Abstract

A graph G is a support for a hypergraph H = (V, S) if the vertices of G correspond to the vertices of H such that for each hyperedge Si ¿ S the subgraph of G induced by Si is connected. G is a planar support if it is a support and planar. Johnson and Pollak [11] proved that it is NPcomplete to decide if a given hypergraph has a planar support. In contrast, there are lienar time algorithms to test whether a given hypergraph has a planar support that is a path, cycle, or tree. In this paper we present an e¿cient algorithm which tests in polynomial time if a given hypergraph has a planar support that is a tree where the maximal degree of each vertex is bounded. Our algorithm is constructive and computes a support if it exists. Furthermore, we prove that it is already NP-hard to decide if a hypergraph has a 2-outerplanar support.
－ BibTeX
```
@article{on-planar-supports-for-hypergraphs:2011,
    title = {On Planar Supports for Hypergraphs},
    author = {K. Buchin and M.J. Kreveld, van and H. Meijer and B. Speckmann and K.A.B. Verbeek},
    year = {2011},
    bookTitle = {Journal of Graph Algorithms and Applications},
}
```
[ PDF ]
A New Method for Subdivision Simplification with Applications to Urban-Area Generalization
K. Buchin, W. Meulemans, and B. Speckmann.
19th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM GIS), pp. 261—270, 2011.

－ Abstract

We introduce a local operation for polygons and subdivisions called an edge-move. Edge-moves do not change the edge orientations present in the input and are thus suitable for iterative simplification or even schematization. Based on edge-moves we present a new efficient method for area- and topology-preserving subdivision simplification. We show how to tailor this generic method towards the specific needs of building wall squaring and urban-area generalization. Our algorithm is guaranteed to make further progress on any subdivision that has two or more faces and/or reflex vertices. Furthermore, our method produces output of high visual quality and is able to generalize maps with approximately 1.8 million edges in a few hours.
－ BibTeX
```
@article{a-new-method-for-subdivision-simplification-with-applications-to-urban-area-generalization:2011,
    title = {A New Method for Subdivision Simplification with Applications to Urban-Area Generalization},
    author = {K. Buchin and W. Meulemans and B. Speckmann},
    year = {2011},
    bookTitle = {19th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM GIS)},
}
```
[ PDF ]
A Splitting Line Model for Directional Relations
K. Buchin, V.J.J. Kusters, B. Speckmann, F. Staals, and B.N. Vasilescu.
Proceedings 19th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM-GIS 2011, Chicago IL, USA, November 1-4, 2011), pp. 142—151, 2011.

－ Abstract

Directional relations are fundamental to spatial data queries, analysis and reasoning. Consequently there has been a significant amount of effort to determine directional relations between two regions. However, many existing methods do not perform well when the regions are neighboring or intertwined. In this paper we introduce a new model for directional relations which is based on a splitting line separating the two regions in question. We identify essential quality criteria for directional relation models and translate them into measurable properties of a given splitting line. We present an efficient algorithm that computes an optimal splitting line for two regions and perform extensive experiments. Our results show that the splitting line model captures directional relations very well and that it clearly outperforms existing approaches on pairs of neighboring or intertwined regions.
－ BibTeX
```
@article{a-splitting-line-model-for-directional-relations:2011,
    title = {A Splitting Line Model for Directional Relations},
    author = {K. Buchin and V.J.J. Kusters and B. Speckmann and F. Staals and B.N. Vasilescu},
    year = {2011},
    bookTitle = {Proceedings 19th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM-GIS 2011, Chicago IL, USA, November 1-4, 2011)},
}
```
Angle-Restricted Steiner Arborescences for Flow Map Layout
K. Buchin, B. Speckmann, and K.A.B. Verbeek.
Algorithms and Computation (22nd International Symposium, ISAAC 2011, Yokohama, Japan, December 5-8, 2011. Proceedings), pp. 250—259, 2011.

－ Abstract

We introduce a new variant of the geometric Steiner arborescence problem, motivated by the layout of flow maps. Flow maps show the movement of objects between places. They reduce visual clutter by bundling lines smoothly and avoiding self-intersections. To capture these properties, our angle-restricted Steiner arborescences, or flux trees, connect several targets to a source with a tree of minimal length whose arcs obey a certain restriction on the angle they form with the source. We study the properties of optimal flux trees and show that they are planar and consist of logarithmic spirals and straight lines. Flux trees have the shallow-light property. Computing optimal flux trees is NP-hard. Hence we consider a variant of flux trees which uses only logarithmic spirals. Spiral trees approximate flux trees within a factor depending on the angle restriction. Computing optimal spiral trees remains NP-hard, but we present an efficient 2-approximation, which can be extended to avoid "positive monotone" obstacles.
－ BibTeX
```
@article{angle-restricted-steiner-arborescences-for-flow-map-layout:2011,
    title = {Angle-Restricted Steiner Arborescences for Flow Map Layout},
    author = {K. Buchin and B. Speckmann and K.A.B. Verbeek},
    year = {2011},
    bookTitle = {Algorithms and Computation (22nd International Symposium, ISAAC 2011, Yokohama, Japan, December 5-8, 2011. Proceedings)},
}
```
Delineating Imprecise Regions Via Shortest-Path Graphs
M.T. Berg, de, W. Meulemans, and B. Speckmann.
19th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM GIS), pp. 271—280, 2011.

－ Abstract

An imprecise region, also called a vernacular region, is a region without a precise or administrative boundary. We present a new method to delineate imprecise regions from a set of points that are likely to lie inside the region. We use shortest-path graphs based on the squared Euclidean distance which capture the shape of region boundaries well. Shortest-path graphs naturally adapt to point sets of varying density, and they are always connected. As opposed to neighborhood graphs, they use a non-local criterion to determine which points to connect. Furthermore, shortest- path graphs can easily be extended to take geographic context into account by modeling context as "soft" obstacles. We present efficient algorithms to compute shortest-path graphs with or without geographic context. We experimentally evaluate the quality of the imprecise regions computed with our method. To fairly compare our results to those obtained by the common KDE approach, we also show how to integrate context into KDE by again using soft obstacles.
－ BibTeX
```
@article{delineating-imprecise-regions-via-shortest-path-graphs:2011,
    title = {Delineating Imprecise Regions Via Shortest-Path Graphs},
    author = {M.T. Berg, de and W. Meulemans and B. Speckmann},
    year = {2011},
    bookTitle = {19th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM GIS)},
}
```
[ PDF ]
Kinetic Convex Hulls and Delaunay Triangulations in the Black-Box Model
M.T. Berg, de, M.J.M. Roeloffzen, and B. Speckmann.
Proceedings 27th Annual ACM Symposium on Computational Geometry (SoCG'11, Paris, France, June 13-15, 2011), pp. 244—253, 2011.

－ Abstract

Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum displacement of any point in one time step. We study the maintenance of the convex hull and the Delaunay triangulation of a planar point set P in the black-box model, under the following assumption on dmax: there is some constant k such that for any point p ¿ P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ¿k , the so-called k-spread of P. We show how to update the convex hull at each time step in O(k¿k log2 n) amortized time. For the Delaunay triangulation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k2 ¿k2) at each time step.
－ BibTeX
```
@article{kinetic-convex-hulls-and-delaunay-triangulations-in-the-black-box-model:2011,
    title = {Kinetic Convex Hulls and Delaunay Triangulations in the Black-Box Model},
    author = {M.T. Berg, de and M.J.M. Roeloffzen and B. Speckmann},
    year = {2011},
    bookTitle = {Proceedings 27th Annual ACM Symposium on Computational Geometry (SoCG'11, Paris, France, June 13-15, 2011)},
}
```
[ PDF ]
Optimizing Regular Edge Labelings
K. Buchin, B. Speckmann, and S. Verdonschot.
Graph Drawing (18th International Symposium, GD'10, Konstanz, Germany, September 21-24, 2010. Revised selected papers), pp. 117—128, 2011.

－ Abstract

A regular edge labeling (REL) of an irreducible triangulation G uniquely defines a rectangular dual of G. Rectangular duals find applications in various areas: as floor plans of electronic chips, in architectural designs, as rectangular cartograms, or as treemaps. An irreducible triangulation can have many RELs and hence many rectangular duals. Depending on the specific application different duals might be desirable. In this paper we consider optimization problems on RELs and show how to find optimal or near-optimal RELs for various quality criteria. Furthermore, we give upper and lower bounds on the number of RELs.
－ BibTeX
```
@article{optimizing-regular-edge-labelings:2011,
    title = {Optimizing Regular Edge Labelings},
    author = {K. Buchin and B. Speckmann and S. Verdonschot},
    year = {2011},
    bookTitle = {Graph Drawing (18th International Symposium, GD'10, Konstanz, Germany, September 21-24, 2010. Revised selected papers)},
}
```
The 2x2 Simple Packing Problem
A.M. Renssen, van and B. Speckmann.
Proceedings of the 23rd Annual Canadian Conference on Computational Geometry (CCCG 2011), August 10-12, 2011, Toronto, Ontario, Canada, pp. 387—392, 2011.

－ Abstract

We significantly extend the class of polygons for which the 22 simple packing problem can be solved in polynomial time.
－ BibTeX
```
@article{the-2x2-simple-packing-problem:2011,
    title = {The 2x2 Simple Packing Problem},
    author = {A.M. Renssen, van and B. Speckmann},
    year = {2011},
    bookTitle = {Proceedings of the 23rd Annual Canadian Conference on Computational Geometry (CCCG 2011), August 10-12, 2011, Toronto, Ontario, Canada},
}
```
[ PDF ]
Treemaps with Bounded Aspect Ratio
M.T. Berg, de, B. Speckmann, and V. Weele, van der.
Algorithms and Computation (22nd International Symposium, ISAAC 2011, Yokohama, Japan, December 5-8, 2011. Proceedings), pp. 260—270, 2011.

－ Abstract

Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree where the weight of each node is the sum of the weights of its children. A treemap for is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of and its area is proportional to the weight of the corresponding node. An important quality criterion for treemaps is the aspect ratio of its regions. One cannot bound the aspect ratio if the regions are restricted to be rectangles. In contrast, polygonal partitions, that use convex polygons, can have bounded aspect ratio. We are the first to obtain convex partitions with optimal aspect ratio O(depth()). However, depth() still depends on the input tree. Hence we introduce a new type of treemaps, namely orthoconvex treemaps, where regions representing leaves are rectangles, L-, and S-shapes, and regions representing internal nodes are orthoconvex polygons. We prove that any input tree, irrespective of the weights of the nodes and the depth of the tree, admits an orthoconvex treemap of constant aspect ratio.
－ BibTeX
```
@article{treemaps-with-bounded-aspect-ratio:2011,
    title = {Treemaps with Bounded Aspect Ratio},
    author = {M.T. Berg, de and B. Speckmann and V. Weele, van der},
    year = {2011},
    bookTitle = {Algorithms and Computation (22nd International Symposium, ISAAC 2011, Yokohama, Japan, December 5-8, 2011. Proceedings)},
}
```
Angle-Restricted Steiner Arborescences for Flow Map Layout
K. Buchin, B. Speckmann, and K.A.B. Verbeek.
pp. 35—38, 2011.

－ Abstract

Flow maps visualize the movement of objects between places. One or more sources are connected to several targets by arcs whose thickness corresponds to the amount of flow between a source and a target. Flow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding self-intersections. We present algorithms that compute crossing-free flows of high visual quality. To this end we introduce a new variant of the geometric Steiner arborescence problem. The goal is to connect the targets to a source with a tree of minimal length whose arcs obey a certain restriction on the angle they form with the source. Such an angle-restricted Steiner arborescence, or simply ow tree, naturally induces a clustering on the targets and smoothly bundles arcs. We study the properties of optimal flow trees and show that they consist of logarithmic spirals and straight lines. Computing optimal flow trees is NP-hard. Hence we consider a variant of flow trees which uses only logarithmic spirals, so called spiral trees. Spiral trees approximate flow trees within a factor depending on the angle restriction. Computing optimal spiral trees remains NP-hard. We present an efficient 2-approximation for spiral trees, which can be extended to avoid certain types of obstacles.
－ BibTeX
```
@article{angle-restricted-steiner-arborescences-for-flow-map-layout:2011,
    title = {Angle-Restricted Steiner Arborescences for Flow Map Layout},
    author = {K. Buchin and B. Speckmann and K.A.B. Verbeek},
    year = {2011},
    bookTitle = {},
}
```
[ PDF ]
Area-Preserving c-Oriented Schematization
K. Buchin, W. Meulemans, and B. Speckmann.
pp. 163—166, 2011.

－ Abstract

We define an edge-move operation for polygons and prove that every simple non-convex polygon P has a non-conflicting pair of complementary edge-moves that reduces the number of edges of P while preserving its area. We use this result to generate area-preserving C-oriented schematizations of polygons.
－ BibTeX
```
@article{area-preserving-c-oriented-schematization:2011,
    title = {Area-Preserving c-Oriented Schematization},
    author = {K. Buchin and W. Meulemans and B. Speckmann},
    year = {2011},
    bookTitle = {},
}
```
[ PDF ]
Convex Treemaps with Bounded Aspect Ratio
M.T. de Berg, B. Speckmann, and V. van der Weele.
pp. 71—74, 2011.

－ Abstract

Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree T where the weight of each node is the sum of the weights of its children. A treemap for T is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of T and the area of each region is proportional to the weight of the corresponding node. An important quality criterium for treemaps is the aspect ratio of its regions. Unfortunately, one cannot bound the aspect ratio if the regions are restricted to be rectangles. Hence Onak and Sidiropoulos introduced polygonal partitions, which use convex polygons. We are the first to obtain convex partitions with optimal aspect ratio O(depth(T )). We also consider the important special case that depth(T ) = 1, that is, single-level treemaps. We show how to construct convex single-level treemaps that use only four simple shapes for the regions and have aspect ratio at most 34/7.
－ BibTeX
```
@article{convex-treemaps-with-bounded-aspect-ratio:2011,
    title = {Convex Treemaps with Bounded Aspect Ratio},
    author = {M.T. de Berg and B. Speckmann and V. van der Weele},
    year = {2011},
    bookTitle = {},
}
```
[ PDF ]
Kinetic Convex Hulls in the Black-Box Model
M.T. de Berg, M.J.M. Roeloffzen, and B. Speckmann.
pp. 201—204, 2011.

－ Abstract

Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on d_max, the maximum displacement of any point in one time step. We study the maintenance of the convex hull of a planar point set P in the black-box model, under the following assumption on d_max: there is some constant k such that for any point p in P the disk of radius d_max contains at most k points. We analyze our algorithms in terms of \Delta_k, the so-called k-spread of P. We show how to update the convex hull at each time step in O(k \Delta_k log^2 n) amortized time.
－ BibTeX
```
@article{kinetic-convex-hulls-in-the-black-box-model:2011,
    title = {Kinetic Convex Hulls in the Black-Box Model},
    author = {M.T. de Berg and M.J.M. Roeloffzen and B. Speckmann},
    year = {2011},
    bookTitle = {},
}
```
[ PDF ]

2010

Algorithmic Aspects of Proportional Symbol Maps
S. Cabello, H.J. Haverkort, M.J. Kreveld, van, and B. Speckmann.
Algorithmica, 58(3):543—565, 2010.

－ Abstract

Proportional symbol maps visualize numerical data associated with point locations by placing a scaled symbol—typically opaque disks or squares—at the corresponding point on a map. Overlapping symbols need to be drawn in such a way that the user can still judge their relative sizes accurately. We identify two types of suitable drawings: physically realizable drawings and stacking drawings. For these we study the following two problems: Max-Min—maximize the minimum visible boundary length of each symbol—and Max-Total—maximize the total visible boundary length over all symbols. We show that both problems are NP-hard for physically realizable drawings. Max-Min can be solved in O(n2 log n) time for stacking drawings, which can be improved to O(n log n) or O(n log2 n) time when the input has certain properties. We also experimented with four methods to compute stacking drawings: our solution to the Max-Min problem performs best on the data sets considered.
－ BibTeX
```
@article{algorithmic-aspects-of-proportional-symbol-maps:2010,
    title = {Algorithmic Aspects of Proportional Symbol Maps},
    author = {S. Cabello and H.J. Haverkort and M.J. Kreveld, van and B. Speckmann},
    year = {2010},
    bookTitle = {Algorithmica},
}
```
Finding the Most Relevant Fragments in Networks
K. Buchin, S. Cabello, J. Gudmundsson, M. Löffler, J. Luo, G. Rote, R.I. Silveira, B. Speckmann, and T. Wolle.
Journal of Graph Algorithms and Applications, 14(2):307—336, 2010.

－ Abstract

We study a point pattern detection problem on networks, motivated by applications in geographical analysis, such as crime hotspot detection. Given a network N (a connected graph with non-negative edge lengths) together with a set of sites, which lie on the edges or vertices of N, we look for a connected subnetwork F of N of small total length that contains many sites. The edges of F can form parts of the edges of N. We consider different variants of this problem where N is either a general graph or restricted to a tree, and the subnetwork F that we are looking for is either a simple path or a tree. We give polynomial-time algorithms, NP-hardness and NP-completeness proofs, approximation algorithms, and also fixed-parameter tractable algorithms.
－ BibTeX
```
@article{finding-the-most-relevant-fragments-in-networks:2010,
    title = {Finding the Most Relevant Fragments in Networks},
    author = {K. Buchin and S. Cabello and J. Gudmundsson and M. Löffler and J. Luo and G. Rote and R.I. Silveira and B. Speckmann and T. Wolle},
    year = {2010},
    bookTitle = {Journal of Graph Algorithms and Applications},
}
```
[ PDF ]
Necklace Maps
B. Speckmann and K.A.B. Verbeek.
IEEE Transactions on Visualization and Computer Graphics, 16(6):881—889, 2010.

－ Abstract

Statistical data associated with geographic regions is nowadays globally available in large amounts and hence automated methods to visually display these data are in high demand. There are several well-established thematic map types for quantitative data on the ratio-scale associated with regions: choropleth maps, cartograms, and proportional symbol maps. However, all these maps suffer from limitations, especially if large data values are associated with small regions. To overcome these limitations, we propose a novel type of quantitative thematic map, the necklace map. In a necklace map, the regions of the underlying two-dimensional map are projected onto intervals on a one-dimensional curve (the necklace) that surrounds the map regions. Symbols are scaled such that their area corresponds to the data of their region and placed without overlap inside the corresponding interval on the necklace. Necklace maps appear clear and uncluttered and allow for comparatively large symbol sizes. They visualize data sets well which are not proportional to region sizes. The linear ordering of the symbols along the necklace facilitates an easy comparison of symbol sizes. One map can contain several nested or disjoint necklaces to visualize clustered data. The advantages of necklace maps come at a price: the association between a symbol and its region is weaker than with other types of maps. Interactivity can help to strengthen this association if necessary. We present an automated approach to generate necklace maps which allows the user to interactively control the final symbol placement. We validate our approach with experiments using various data sets and maps.
－ BibTeX
```
@article{necklace-maps:2010,
    title = {Necklace Maps},
    author = {B. Speckmann and K.A.B. Verbeek},
    year = {2010},
    bookTitle = {IEEE Transactions on Visualization and Computer Graphics},
}
```
[ PDF ]
Optimal BSPs and Rectilinear Cartograms
M. Berg, de, E. Mumford, and B. Speckmann.
International Journal of Computational Geometry and Applications, 20(2):203—222, 2010.

－ Abstract

A cartogram is a thematic map that visualizes statistical data about a set of regions like countries, states or provinces. The size of a region in a cartogram corresponds to a particular geographic variable, for example, population. We present an algorithm for constructing rectilinear cartograms (each region is represented by a rectilinear polygon) with zero cartographic error and correct region adjacencies, and we test our algorithm on various data sets. It produces regions of very small complexity—in fact, most regions are rectangles—while still ensuring both exact areas and correct adjacencies for all regions. Our algorithm uses a novel subroutine that is interesting in its own right, namely a polynomial-time algorithm for computing optimal binary space partitions (BSPs) for rectilinear maps. This algorithm works for a general class of optimality criteria, including size and depth. We use this generality in our application to computing cartograms, where we apply a dedicated cost function leading to BSPs amenable to the constructing of high-quality cartograms.
－ BibTeX
```
@article{optimal-bsps-and-rectilinear-cartograms:2010,
    title = {Optimal BSPs and Rectilinear Cartograms},
    author = {M. Berg, de and E. Mumford and B. Speckmann},
    year = {2010},
    bookTitle = {International Journal of Computational Geometry and Applications},
}
```
Pointed Binary Encompassing Trees: Simple and Optimal
M. Hoffmann, B. Speckmann, and Cs.D. Tóth.
Computational Geometry, 43(1):35—41, 2010.

－ Abstract

For n disjoint line segments in the plane we construct in optimal O(nlogn) time and linear space an encompassing tree of maximum degree three such that at every vertex v all edges of the tree that are incident to v lie in a halfplane bounded by the line through the input segment which v is an endpoint of. In particular, this tree is pointed since every vertex has an incident angle greater than p. Such a pointed binary tree can be augmented to a minimum pseudo-triangulation. It follows that every set of disjoint line segments in the plane has a constrained minimum pseudo-triangulation whose maximum vertex degree is bounded by a constant.
－ BibTeX
```
@article{pointed-binary-encompassing-trees-simple-and-optimal:2010,
    title = {Pointed Binary Encompassing Trees: Simple and Optimal},
    author = {M. Hoffmann and B. Speckmann and Cs.D. Tóth},
    year = {2010},
    bookTitle = {Computational Geometry},
}
```
Area-Preserving Subdivision Schematization
W. Meulemans, A.M. Renssen, van, and B. Speckmann.
6th International Conference on Geographic Information Science (GIScience), pp. 160—174, 2010.

－ Abstract

We describe an area-preserving subdivision schematization algorithm: the area of each region in the input equals the area of the corresponding region in the output. Our schematization is axis-aligned, the final output is a rectilinear subdivision. We first describe how to convert a given subdivision into an area-equivalent rectilinear subdivision. Then we define two area-preserving contraction operations and prove that at least one of these operations can always be applied to any given simple rectilinear polygon. We extend this approach to subdivisions and showcase experimental results. Finally, we give examples for standard distance metrics (symmetric difference, Hausdorff- and Fréchet-distance) that show that better schematizations might result in worse shapes.
－ BibTeX
```
@article{area-preserving-subdivision-schematization:2010,
    title = {Area-Preserving Subdivision Schematization},
    author = {W. Meulemans and A.M. Renssen, van and B. Speckmann},
    year = {2010},
    bookTitle = {6th International Conference on Geographic Information Science (GIScience)},
}
```
[ PDF ]
Geometric Simultaneous Embeddings of a Graph and a Matching
S. Cabello, M.J. Kreveld, van, G. Liotta, H. Meijer, B. Speckmann, and K.A.B. Verbeek.
Graph Drawing (17th International Symposium, GD'09, Chicago, IL, USA, September 22-25, 2009. Revised Papers), pp. 183—194, 2010.

－ Abstract

The geometric simultaneous embedding problem asks whether two planar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous embedding is a current topic in graph drawing and positive and negative results are known for various classes of graphs. So far only connected graphs have been considered. In this paper we present the first results for the setting where one of the graphs is a matching. In particular, we show that there exists a planar graph and a matching which do not admit a geometric simultaneous embedding. This generalizes the same result for a planar graph and a path. On the positive side, we describe algorithms that compute a geometric simultaneous embedding of a matching and a wheel, outerpath, or tree. Our proof for a matching and a tree sheds new light on a major open question: do a tree and a path always admit a geometric simultaneous embedding? Our drawing algorithms minimize the number of orientations used to draw the edges of the matching. Specifically, when embedding a matching and a tree, we can draw all matching edges horizontally. When embedding a matching and a wheel or an outerpath, we use only two orientations.
－ BibTeX
```
@article{geometric-simultaneous-embeddings-of-a-graph-and-a-matching:2010,
    title = {Geometric Simultaneous Embeddings of a Graph and a Matching},
    author = {S. Cabello and M.J. Kreveld, van and G. Liotta and H. Meijer and B. Speckmann and K.A.B. Verbeek},
    year = {2010},
    bookTitle = {Graph Drawing (17th International Symposium, GD'09, Chicago, IL, USA, September 22-25, 2009. Revised Papers)},
}
```
Homotopic Rectilinear Routing with Few Links and Thick Edges
B. Speckmann and K.A.B. Verbeek.
LATIN 2010: Theoretical Informatics (Proceedings 9th Latin American Symposium, Oaxaca, Mexico, April 19-23, 2010), pp. 468—479, 2010.

－ Abstract

We study the NP-hard problem of finding non-crossing thick minimum-link rectilinear paths which are homotopic to a set of input paths in an environment with rectangular obstacles. We present a 2-approximation that runs in O(n 3 + k_in log n+ k_out) time, where n is the total number of input paths and obstacles and k in and k out are the total complexities of the input and output paths. Our algorithm not only approximates the minimum number of links, but also simultaneously minimizes the total length of the paths. We also show that an approximation factor of 2 is optimal when using smallest paths as lower bound.
－ BibTeX
```
@article{homotopic-rectilinear-routing-with-few-links-and-thick-edges:2010,
    title = {Homotopic Rectilinear Routing with Few Links and Thick Edges},
    author = {B. Speckmann and K.A.B. Verbeek},
    year = {2010},
    bookTitle = {LATIN 2010: Theoretical Informatics (Proceedings 9th Latin American Symposium, Oaxaca, Mexico, April 19-23, 2010)},
}
```
On Planar Supports for Hypergraphs
K. Buchin, M.J. Kreveld, van, H. Meijer, B. Speckmann, and K.A.B. Verbeek.
Graph Drawing (17th International Symposium, GD'09, Chicago, IL, USA, September 22-25, 2009. Revised Papers), pp. 345—356, 2010.

－ Abstract

A graph G is a support for a hypergraph H=(V,S) if the vertices of G correspond to the vertices of H such that for each hyperedge Si e S the subgraph of G induced by Si is connected. G is a planar support if it is a support and planar. Johnson and Pollak [9] proved that it is NP-complete to decide if a given hypergraph has a planar support. In contrast, there are polynomial time algorithms to test whether a given hypergraph has a planar support that is a path, cycle, or tree. In this paper we present an algorithm which tests in polynomial time if a given hypergraph has a planar support that is a tree where the maximal degree of each vertex is bounded. Our algorithm is constructive and computes a support if it exists. Furthermore, we prove that it is already NP-hard to decide if a hypergraph has a 3-outerplanar support.
－ BibTeX
```
@article{on-planar-supports-for-hypergraphs:2010,
    title = {On Planar Supports for Hypergraphs},
    author = {K. Buchin and M.J. Kreveld, van and H. Meijer and B. Speckmann and K.A.B. Verbeek},
    year = {2010},
    bookTitle = {Graph Drawing (17th International Symposium, GD'09, Chicago, IL, USA, September 22-25, 2009. Revised Papers)},
}
```
Treemaps with Bounded Aspect Ratio
M.T. Berg, de, B. Speckmann, and V. Weele, van der.
2010.

－ Abstract

Treemaps are a popular technique to visualize hierarchical data. The input is a weighted tree $\tree$ where the weight of each node is the sum of the weights of its children. A treemap for $\tree$ is a hierarchical partition of a rectangle into simply connected regions, usually rectangles. Each region represents a node of $\tree$ and the area of each region is proportional to the weight of the corresponding node. An important quality criterium for treemaps is the aspect ratio of its regions. Unfortunately, one cannot bound the aspect ratio if the regions are restricted to be rectangles. Hence Onak and Sidiropoulos in SoCG 2008 introduced \emph{polygonal partitions}, which use convex polygons. We are the first to obtain convex partitions with optimal aspect ratio $O(\depth(\tree))$. Furthermore, we consider rectilinear partitions, which retain more of the schematized flavor of standard rectangular treemaps. Our rectilinear treemaps have constant aspect ratio, independent of $\depth(\tree)$ or the number and weight of the nodes. The leaves of $\tree$ are represented by rectangles, L-, and S-shapes and internal nodes by orthoconvex polygons. We also consider the important special case that $\depth(\tree)=1$, that is, single-level treemaps. We prove that it is strongly NP-hard to minimize the aspect ratio of a rectangular single-level treemap. On the positive side we show how to construct rectilinear and convex single-level treemaps with constant aspect ratio. Our rectilinear single-level treemaps use only rectangles and L-shapes and have aspect ratio at most $2 + 2 \sqrt{3}/3$. The convex version uses four different octilinear shapes and has aspect ratio at most 9/2.
－ BibTeX
```
@article{treemaps-with-bounded-aspect-ratio:2010,
    title = {Treemaps with Bounded Aspect Ratio},
    author = {M.T. Berg, de and B. Speckmann and V. Weele, van der},
    year = {2010},
    bookTitle = {},
}
```

2009

Edges and Switches, Tunnels and Bridges
D. Eppstein, M.J. Kreveld, van, E. Mumford, and B. Speckmann.
Computational Geometry, 42(8):790—802, 2009.

－ Abstract

Edge casing is a well-known method to improve the readability of drawings of non-planar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than others. We formulate several optimization criteria that try to capture the concept of a "good" cased drawing. Further, we address the algorithmic question of how to turn a given drawing into an optimal cased drawing. For many of the resulting optimization problems, we either find polynomial time algorithms or NP-hardness results.
－ BibTeX
```
@article{edges-and-switches-tunnels-and-bridges:2009,
    title = {Edges and Switches, Tunnels and Bridges},
    author = {D. Eppstein and M.J. Kreveld, van and E. Mumford and B. Speckmann},
    year = {2009},
    bookTitle = {Computational Geometry},
}
```
Flip Graphs of Bounded-Degree Triangulations
O. Aichholzer, T. Hackl, D. Orden, P. Ramos, G. Rote, A. Schulz, and B. Speckmann.
Electronic Notes in Discrete Mathematics, 34:509—513, 2009.

－ Abstract

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant k. Specifically, we consider triangulations of sets of n points in convex position in the plane and prove that their flip graph is connected if and only if k>6; the diameter of the flip graph is O(n2). We also show that for general point sets, flip graphs of triangulations with degree k can be disconnected for any k.
－ BibTeX
```
@article{flip-graphs-of-bounded-degree-triangulations:2009,
    title = {Flip Graphs of Bounded-Degree Triangulations},
    author = {O. Aichholzer and T. Hackl and D. Orden and P. Ramos and G. Rote and A. Schulz and B. Speckmann},
    year = {2009},
    bookTitle = {Electronic Notes in Discrete Mathematics},
}
```
Kinetic Collision Detection for Convex Fat Objects
M.A. Abam, M. Berg, de, S.H. Poon, and B. Speckmann.
Algorithmica, 53(4):457—473, 2009.

－ Abstract

We design compact and responsive kinetic data structures for detecting collisions between n convex fat objects in 3-dimensional space that can have arbitrary sizes. Our main results are: (i) If the objects are 3-dimensional balls that roll on a plane, then we can detect collisions with a KDS of size O(nlog¿n) that can handle events in O(log¿2 n) time. This structure processes O(n 2) events in the worst case, assuming that the objects follow constant-degree algebraic trajectories. (ii) If the objects are convex fat 3-dimensional objects of constant complexity that are free-flying in R3, then we can detect collisions with a KDS of O(nlog¿6 n) size that can handle events in O(log¿7 n) time. This structure processes O(n 2) events in the worst case, assuming that the objects follow constant-degree algebraic trajectories. If the objects have similar sizes then the size of the KDS becomes O(n) and events can be handled in O(log¿n) time.
－ BibTeX
```
@article{kinetic-collision-detection-for-convex-fat-objects:2009,
    title = {Kinetic Collision Detection for Convex Fat Objects},
    author = {M.A. Abam and M. Berg, de and S.H. Poon and B. Speckmann},
    year = {2009},
    bookTitle = {Algorithmica},
}
```
Kinetic Kd-Trees and Longest-Side Kd-Trees
M.A. Abam, M. Berg, de, and B. Speckmann.
SIAM Journal on Computing, 39(4):1219—1232, 2009.

－ Abstract

We propose a simple variant of kd-trees, called rank-based kd-trees, for sets of n points in Rd. We show that a rank-based kd-tree, like an ordinary kd-tree, supports orthogonal range queries in O(n1-1/d + k) time, where k is the output size. The main advantage of rank-based kd-trees is that they can be efficiently kinetized: the kinetic data structure (KDS) processes O(n2) events in the worst case, assuming that the points follow constant-degree algebraic trajectories; each event can be handled in O(log n) time, and each point is involved in O(1) certificates. We also propose a variant of longest-side kd-trees, called rank-based longest-side kd-trees, for sets of points in R2. Rank-based longest-side kd-trees can be kinetized efficiently as well, and like longest-side kd-trees, they support e-approximate nearest-neighbor, e-approximate farthest-neighbor, and e-approximate range queries with convex ranges in O((1/e) log2 n) time. The KDS processes O(n3 log n) events in the worst case, assuming that the points follow constant-degree algebraic trajectories; each event can be handled in O(log2 n) time, and each point is involved in O(log n) certificates.
－ BibTeX
```
@article{kinetic-kd-trees-and-longest-side-kd-trees:2009,
    title = {Kinetic Kd-Trees and Longest-Side Kd-Trees},
    author = {M.A. Abam and M. Berg, de and B. Speckmann},
    year = {2009},
    bookTitle = {SIAM Journal on Computing},
}
```
[ PDF ]
Matched Drawings of Planar Graphs
E. Di Giacomo, W. Didimo, M.J. Kreveld, van, G. Liotta, and B. Speckmann.
Journal of Graph Algorithms and Applications, 13(3):423—445, 2009.

－ Abstract

A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which is a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3-connected planar graphs or a 3-connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a sufficiently restricted planar grap - such as an unlabeled level planar (ULP) graph or a graph of the family of "carousel graphs" - are always matched drawable.
－ BibTeX
```
@article{matched-drawings-of-planar-graphs:2009,
    title = {Matched Drawings of Planar Graphs},
    author = {E. Di Giacomo and W. Didimo and M.J. Kreveld, van and G. Liotta and B. Speckmann},
    year = {2009},
    bookTitle = {Journal of Graph Algorithms and Applications},
}
```
[ PDF ]
On Minimum Weight Pseudo-Triangulations
O. Aichholzer, F. Aurenhammer, T. Hackl, and B. Speckmann.
Computational Geometry, 42(6-7):627—631, 2009.

－ Abstract

In this note we discuss some structural properties of minimum weight pseudo-triangulations of point sets.
－ BibTeX
```
@article{on-minimum-weight-pseudo-triangulations:2009,
    title = {On Minimum Weight Pseudo-Triangulations},
    author = {O. Aichholzer and F. Aurenhammer and T. Hackl and B. Speckmann},
    year = {2009},
    bookTitle = {Computational Geometry},
}
```
On Rectilinear Duals for Vertex-Weighted Plane Graphs
M. Berg, de, E. Mumford, and B. Speckmann.
Discrete Mathematics, 309(7):1794—1812, 2009.

－ Abstract

Let G= (V, E) be a plane triangulated graph where each vertex is assigned a positive weight. A rectilinear dual of is a partition of a rectangle into |V| simple rectilinear regions, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge in E. A rectilinear dual is called a cartogram if the area of each region is equal to the weight of the corresponding vertex. We show that every vertex-weighted plane triangulated graph admits a cartogram of constant complexity, that is, a cartogram where the number of vertices of each region is constant. Furthermore, such a rectilinear cartogram can be constructed in O(nlogn) time where n=|V|.
－ BibTeX
```
@article{on-rectilinear-duals-for-vertex-weighted-plane-graphs:2009,
    title = {On Rectilinear Duals for Vertex-Weighted Plane Graphs},
    author = {M. Berg, de and E. Mumford and B. Speckmann},
    year = {2009},
    bookTitle = {Discrete Mathematics},
}
```
Polychromatic Colorings of Plane Graphs
N. Alon, R. Berke, K. Buchin, M. Buchin, P. Csorba, S. Shannigrahi, B. Speckmann, and P. Zumstein.
Discrete and Computational Geometry, 42(3):421—442, 2009.

－ Abstract

We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by ¿(3g-5)/4¿ colors so that every color appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and that admit no vertex coloring of this type with more than ¿(3g+1)/4¿ colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by k colors in which all colors appear in every face is in P for k=2 and is -complete for k=3,4. We refine this result for polychromatic 3-colorings restricted to 2-connected graphs which have face sizes from a prescribed (possibly infinite) set of integers. Thereby we find an almost complete characterization of these sets of integers (face sizes) for which the corresponding decision problem is in P, and for the others it is -complete.
－ BibTeX
```
@article{polychromatic-colorings-of-plane-graphs:2009,
    title = {Polychromatic Colorings of Plane Graphs},
    author = {N. Alon and R. Berke and K. Buchin and M. Buchin and P. Csorba and S. Shannigrahi and B. Speckmann and P. Zumstein},
    year = {2009},
    bookTitle = {Discrete and Computational Geometry},
}
```
Area-Universal Rectangular Layouts
D. Eppstein, E. Mumford, B. Speckmann, and K.A.B. Verbeek.
Proceedings 25th Annual ACM Symposium on Computational Geometry (SoCG'09, Aarhus, Denmark, June 8-10, 2009), pp. 267—276, 2009.

－ Abstract

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. They are used as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it is desirable for one rectangular layout to represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.
－ BibTeX
```
@article{area-universal-rectangular-layouts:2009,
    title = {Area-Universal Rectangular Layouts},
    author = {D. Eppstein and E. Mumford and B. Speckmann and K.A.B. Verbeek},
    year = {2009},
    bookTitle = {Proceedings 25th Annual ACM Symposium on Computational Geometry (SoCG'09, Aarhus, Denmark, June 8-10, 2009)},
}
```
Connect the Dot: Computing Feed-Links with Minimum Dilation
B. Aronov, K. Buchin, M. Buchin, M.J. Kreveld, van, M. Löffler, J. Luo, R.I. Silveira, and B. Speckmann.
Algorithms and Data Structures (Proceedings 11th International Workshop, WADS 2009, Banff, Alberta, Canada, August 21-23, 2009), pp. 49—60, 2009.

－ Abstract

A feed-link is an artificial connection from a given location p to a real-world network. It is most commonly added to an incomplete network to improve the results of network analysis, by making p part of the network. The feed-link has to be "reasonable", hence we use the concept of dilation to determine the quality of a connection. We consider the following abstract problem: Given a simple polygon P with n vertices and a point p inside, determine a point q on P such that adding a feedlink minimizes the maximum dilation of any point on P. Here the dilation of a point r on P is the ratio of the shortest route from r over P and to p, to the Euclidean distance from r to p. We solve this problem in O(¿ 7(n)logn) time, where ¿ 7(n) is the slightly superlinear maximum length of a Davenport-Schinzel sequence of order 7. We also show that for convex polygons, two feed-links are always sufficient and sometimes necessary to realize constant dilation, and that k feed-links lead to a dilation of 1¿+¿O(1/k). For (a,ß)-covered polygons, a constant number of feed-links suffices to realize constant dilation.
－ BibTeX
```
@article{connect-the-dot-computing-feed-links-with-minimum-dilation:2009,
    title = {Connect the Dot: Computing Feed-Links with Minimum Dilation},
    author = {B. Aronov and K. Buchin and M. Buchin and M.J. Kreveld, van and M. Löffler and J. Luo and R.I. Silveira and B. Speckmann},
    year = {2009},
    bookTitle = {Algorithms and Data Structures (Proceedings 11th International Workshop, WADS 2009, Banff, Alberta, Canada, August 21-23, 2009)},
}
```
Detecting Hotspots in Geographic Networks
K. Buchin, S. Cabello, J. Gudmundsson, M. Löffler, J. Luo, G. Rote, R.I. Silveira, B. Speckmann, and T. Wolle.
Advances in GIScience (Proceedings of the 12th AGILE Conference, Hannover, Germany, June 2-5, 2009), pp. 217—231, 2009.

－ Abstract

We study a point pattern detection problem on networks, motivated by geographical analysis tasks, such as crime hotspot detection. Given a network N (for example, a street, train, or highway network) together with a set of sites which are located on the network (for example, accident locations or crime scenes), we want to find a connected subnetwork F of N of small total length that contains many sites. That is, we are searching for a subnetwork F that spans a cluster of sites which are close with respect to the network distance. We consider different variants of this problem where N is either a general graph or restricted to a tree, and the subnetwork F that we are looking for is either a simple path, a path with self-intersections at vertices, or a tree. Many of these variants are NP-hard, that is, polynomial-time solutions are very unlikely to exist. Hence we focus on exact algorithms for special cases and efficient algorithms for the general case under realistic input assumptions.
－ BibTeX
```
@article{detecting-hotspots-in-geographic-networks:2009,
    title = {Detecting Hotspots in Geographic Networks},
    author = {K. Buchin and S. Cabello and J. Gudmundsson and M. Löffler and J. Luo and G. Rote and R.I. Silveira and B. Speckmann and T. Wolle},
    year = {2009},
    bookTitle = {Advances in GIScience (Proceedings of the 12th AGILE Conference, Hannover, Germany, June 2-5, 2009)},
}
```
Plane Graphs with Parity Constraints
O. Aichholzer, T. Hackl, M. Hoffmann, A. Pilz, G. Rote, B. Speckmann, and B. Vogtenhuber.
Algorithms and Data Structures (Proceedings 11th International Workshop, WADS 2009, Banff, Alberta, Canada, August 21-23, 2009), pp. 13—24, 2009.

－ Abstract

Let S be a set of n points in general position in the plane. Together with S we are given a set of parity constraints, that is, every point of S is labeled either even or odd. A graph G on S satisfies the parity constraint of a point p¿¿¿S, if the parity of the degree of p in G matches its label. In this paper we study how well various classes of planar graphs can satisfy arbitrary parity constraints. Specifically, we show that we can always find a plane tree, a two-connected outerplanar graph, or a pointed pseudo-triangulation which satisfy all but at most three parity constraints. With triangulations we can satisfy about 2/3 of all parity constraints. In contrast, for a given simple polygon H with polygonal holes on S, we show that it is NP-complete to decide whether there exists a triangulation of H that satisfies all parity constraints.
－ BibTeX
```
@article{plane-graphs-with-parity-constraints:2009,
    title = {Plane Graphs with Parity Constraints},
    author = {O. Aichholzer and T. Hackl and M. Hoffmann and A. Pilz and G. Rote and B. Speckmann and B. Vogtenhuber},
    year = {2009},
    bookTitle = {Algorithms and Data Structures (Proceedings 11th International Workshop, WADS 2009, Banff, Alberta, Canada, August 21-23, 2009)},
}
```
Rectangular Cartograms: the Game
M.T. Berg, de, F.S.B. Nijnatten, van, B. Speckmann, and K.A.B. Verbeek.
Proceedings 25th Annual ACM Symposium on Computational Geometry (SoCG'09, Aarhus, Denmark, June 8-10, 2009), pp. 96—97, 2009.

－ Abstract

Raisz [3] introduced rectangular cartograms in 1934 as a way of visualizing spatial information, such as population or economic strength, of a set of regions like countries or states. Rectangular cartograms represent geographic regions by rectangles; the positioning and adjacencies of the rectangles are chosen according to their geographic locations, while their areas are proportional to the numeric values being communicated by the cartogram. Rectangles have the advantage that the sizes (area) of the regions can be estimated easily. On the other hand the rectangular shape is less recognizable and imposes limitations on possible layouts of the cartogram. In recent years several algorithms [2, 4, 5] were developed to efficiently compute rectangular cartograms. This note describes a game that was inspired by these algorithmic results. The game was initially developed as part of the outreach effort of TU Eindhoven, and it has been used at TU Eindhoven’s Open Day. The goal of the Open Day is to popularize the research activities of the various departments; the audience are often families with children in the age 6–15, or parents of (prospective) students. The goal of our game was thus to show in an entertaining way the difficulties one faces when developing algorithms for automatic cartogram construction. The game was quite popular, especially with children. It is an applet and can be found at http://www.win.tue.nl/~speckman/demos/game/. Following [4, 5] the cartograms used in the game are each based on a rectangular layout: a subdivision of a rectangle into finitely many interior-disjoint rectangles. Each region of the input map corresponds to a rectangle of the layout, in addition the layout may contain additional "sea rectangles" that help to preserve the original outline of the regions. See, for example, Figure 1: the lower left corner shows a layout for the currently loaded map, in this case, the provinces of the Netherlands. The layout precisely specifies the required rectangle adjacencies of the final cartogram.
－ BibTeX
```
@article{rectangular-cartograms-the-game:2009,
    title = {Rectangular Cartograms: the Game},
    author = {M.T. Berg, de and F.S.B. Nijnatten, van and B. Speckmann and K.A.B. Verbeek},
    year = {2009},
    bookTitle = {Proceedings 25th Annual ACM Symposium on Computational Geometry (SoCG'09, Aarhus, Denmark, June 8-10, 2009)},
}
```
Shooting Permanent Rays Among Disjoint Polygons in the Plane
M. Ishaque, B. Speckmann, and Cs.D. Tóth.
Proceedings 25th Annual ACM Symposium on Computational Geometry (SoCG'09, Aarhus, Denmark, June 8-10, 2009), pp. 51—60, 2009.

－ Abstract

We present a data structure for ray shooting-and-insertion in the free space among disjoint polygonal obstacles with a total of $n$ vertices in the plane, where each ray starts at the boundary of some obstacle. The portion of each query ray between the starting point and the first obstacle hit is inserted permanently as a new obstacle. Our data structure uses O(n log n) space and preprocessing time, and it supports m successive ray shooting-and-insertion queries in O(n log2 n + m log m) total time. We present two applications for our data structure: (1) Our data structure supports efficient implementation of auto-partitions in the plane i.e. binary space partitions where each partition is done along the supporting line of an input segment. If n input line segments are fragmented into m pieces by an auto-partition, then it can now be implemented in O(n log2n+m log m) time. This improves the expected runtime of Patersen and Yao's classical randomized auto-partition algorithm for n disjoint line segments to O(n log2 n). (2) If we are given disjoint polygonal obstacles with a total of n vertices in the plane, a permutation of the reflex vertices, and a half-line at each reflex vertex that partitions the reflex angle into two convex angles, then the folklore convex partitioning algorithm draws a ray emanating from each reflex vertex in the prescribed order in the given direction until it hits another obstacle, a previous ray, or infinity. The previously best implementation (with a semi-dynamic ray shooting data structure) requires O(n3/2-e/2) time using O(n1+e) space. Our data structure improves the runtime to O(n log2 n).
－ BibTeX
```
@article{shooting-permanent-rays-among-disjoint-polygons-in-the-plane:2009,
    title = {Shooting Permanent Rays Among Disjoint Polygons in the Plane},
    author = {M. Ishaque and B. Speckmann and Cs.D. Tóth},
    year = {2009},
    bookTitle = {Proceedings 25th Annual ACM Symposium on Computational Geometry (SoCG'09, Aarhus, Denmark, June 8-10, 2009)},
}
```
Subdivision Drawings of Hypergraphs
M. Kaufmann, M.J. Kreveld, van, and B. Speckmann.
Graph Drawing (16th International Symposium, GD'08, Heraklion, Crete, Greece, September 21-24, 2008, Revised Papers), pp. 396—407, 2009.

－ Abstract

We introduce the concept of subdivision drawings of hypergraphs. In a subdivision drawing each vertex corresponds uniquely to a face of a planar subdivision and, for each hyperedge, the union of the faces corresponding to the vertices incident to that hyperedge is connected. Vertex-based Venn diagrams and concrete Euler diagrams are both subdivision drawings. In this paper we study two new types of subdivision drawings which are more general than concrete Euler diagrams and more restricted than vertex-based Venn diagrams. They allow us to draw more hypergraphs than the former while having better aesthetic properties than the latter. This research was initiated during the Bertinoro Workshop on Graph Drawing, 2008. Bettina Speckmann is supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.022.707.
－ BibTeX
```
@article{subdivision-drawings-of-hypergraphs:2009,
    title = {Subdivision Drawings of Hypergraphs},
    author = {M. Kaufmann and M.J. Kreveld, van and B. Speckmann},
    year = {2009},
    bookTitle = {Graph Drawing (16th International Symposium, GD'08, Heraklion, Crete, Greece, September 21-24, 2008, Revised Papers)},
}
```
Area-Universal Rectangular Layouts
D. Eppstein, E. Mumford, B. Speckmann, and K.A.B. Verbeek.
pp. 247—250, 2009.

－ Abstract

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. More generally, given any rectangular layout L and any assignment of areas to its regions, we show that there can be at most one layout (up to horizontal and vertical scaling) which is combinatorially equivalent to L and achieves a given area assignment. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.
－ BibTeX
```
@article{area-universal-rectangular-layouts:2009,
    title = {Area-Universal Rectangular Layouts},
    author = {D. Eppstein and E. Mumford and B. Speckmann and K.A.B. Verbeek},
    year = {2009},
    bookTitle = {},
}
```
[ PDF ]
Homotopic Rectilinear Routing with Few Links and Thick Edges
B. Speckmann and K.A.B. Verbeek.
pp. 243—246, 2009.

－ Abstract

We study the problem of finding non-crossing thick minimum-link rectilinear paths homotopic to a set of input paths in an environment with rectangular obstacles. This problem occurs in the context of map schematization under geometric embedding restrictions, for example, when schematizing a highway network for use as a thematic layer. We present a 2-approximation algorithm that runs in O(n3 +kin log n + kout) time, where n is the total number of input paths and obstacles and kin and kout are the total complexities of the input and output paths, respectively. Our algorithm not only approximates the minimum number of links, but also minimizes the total length of the paths. An approximation factor of 2 is optimal when using smallest paths as lower bound.
－ BibTeX
```
@article{homotopic-rectilinear-routing-with-few-links-and-thick-edges:2009,
    title = {Homotopic Rectilinear Routing with Few Links and Thick Edges},
    author = {B. Speckmann and K.A.B. Verbeek},
    year = {2009},
    bookTitle = {},
}
```
[ PDF ]
Area-Universal Rectangular Layouts
D. Eppstein, E. Mumford, B. Speckmann, and K.A.B. Verbeek.
2009.

－ Abstract

A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. Rectangular layouts appear in various applications: as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it might hence be desirable if one rectangular layout can represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. More generally, given any rectangular layout L and any assignment of areas to its regions, we show that there can be at most one layout (up to horizontal and vertical scaling) which is combinatorially equivalent to L and achieves a given area assignment. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists.
－ BibTeX
```
@article{area-universal-rectangular-layouts:2009,
    title = {Area-Universal Rectangular Layouts},
    author = {D. Eppstein and E. Mumford and B. Speckmann and K.A.B. Verbeek},
    year = {2009},
    bookTitle = {},
}
```

On Planar Supports for Hypergraphs
K. Buchin, M.J. Kreveld, van, H. Meijer, B. Speckmann, and K.A.B. Verbeek.
2009.

－ BibTeX

@article{on-planar-supports-for-hypergraphs:2009,
    title = {On Planar Supports for Hypergraphs},
    author = {K. Buchin and M.J. Kreveld, van and H. Meijer and B. Speckmann and K.A.B. Verbeek},
    year = {2009},
    bookTitle = {},
}

2008

Kinetic Data Structures
B. Speckmann.
Encyclopedia of Algorithms, pp. 417—419, 2008.

－ BibTeX

@article{kinetic-data-structures:2008,
    title = {Kinetic Data Structures},
    author = {B. Speckmann},
    year = {2008},
    bookTitle = {Encyclopedia of Algorithms},
}

Efficient Algorithms for Maximum Regression Depth
M.J. Kreveld, van, J.S.B. Mitchell, P.J. Rousseeuw, M. Sharir, J. Snoeyink, and B. Speckmann.
Discrete and Computational Geometry, 39(4):656—677, 2008.

－ Abstract

We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n d ) time and O(n d-1) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(nlog¿2 n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(nlog¿n) time algorithm by Langerman and Steiger (Discrete Comput. Geom. 30(2):299–309, [2003]). Furthermore, we describe the structure of depth in the plane and higher dimensions, leading to various other geometric and algorithmic results. A preliminary version of this paper appeared in the proceedings of the 15th Annual ACM Symposium on Computational Geometry (1999)
－ BibTeX
```
@article{efficient-algorithms-for-maximum-regression-depth:2008,
    title = {Efficient Algorithms for Maximum Regression Depth},
    author = {M.J. Kreveld, van and J.S.B. Mitchell and P.J. Rousseeuw and M. Sharir and J. Snoeyink and B. Speckmann},
    year = {2008},
    bookTitle = {Discrete and Computational Geometry},
}
```
On the Number of Pseudo-Triangulations of Certain Point Sets
O. Aichholzer, D. Orden, F. Santos, and B. Speckmann.
Journal of Combinatorial Theory, Series A, 115(2):254—278, 2008.

－ Abstract

We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically 12n nT(1) pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far.
－ BibTeX
```
@article{on-the-number-of-pseudo-triangulations-of-certain-point-sets:2008,
    title = {On the Number of Pseudo-Triangulations of Certain Point Sets},
    author = {O. Aichholzer and D. Orden and F. Santos and B. Speckmann},
    year = {2008},
    bookTitle = {Journal of Combinatorial Theory, Series A},
}
```
Placing Diagrams and Symbols on Maps
M.J. Kreveld, van and B. Speckmann.
Nieuwsbrief van de Nederlandse Vereniging voor Theoretische Informatica, 12:14—24, 2008.

－ Abstract

The most commonly used types of maps are road maps or topographic maps. However, to convey specialized information, such as statistical data associated with various regions, cartographers use special purpose maps. This note describes some algorithmic questions that arise from the automated generation of two types of these maps, namely maps that contain diagrams and proportional symbol maps. We also sketch some solution strategies and show the output generated by our algorithms.
－ BibTeX
```
@article{placing-diagrams-and-symbols-on-maps:2008,
    title = {Placing Diagrams and Symbols on Maps},
    author = {M.J. Kreveld, van and B. Speckmann},
    year = {2008},
    bookTitle = {Nieuwsbrief van de Nederlandse Vereniging voor Theoretische Informatica},
}
```
Feed-Links for Network Extensions
B. Aronov, K. Buchin, M. Buchin, B.M.P. Jansen, T. Jong, de, M.J. Kreveld, van, M. Löffler, J. Luo, R.I. Silveira, and B. Speckmann.
Proceedings 16th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM-GIS'08, Irvine CA, USA, November 5-7, 2008), pp. 35—1/9, 2008.

－ Abstract

Road network data is often incomplete, making it hard to perform network analysis. This paper discusses the problem of extending partial road networks with reasonable links, using the concept of dilation (also known as crow flight conversion coefficient). To this end, we study how to connect a point (relevant location) inside a polygon (face of the known part of the road network) to the boundary so that the dilation from that point to any point on the boundary is not too large. We provide algorithms and heuristics, and give a computational and experimental analysis.
－ BibTeX
```
@article{feed-links-for-network-extensions:2008,
    title = {Feed-Links for Network Extensions},
    author = {B. Aronov and K. Buchin and M. Buchin and B.M.P. Jansen and T. Jong, de and M.J. Kreveld, van and M. Löffler and J. Luo and R.I. Silveira and B. Speckmann},
    year = {2008},
    bookTitle = {Proceedings 16th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM-GIS'08, Irvine CA, USA, November 5-7, 2008)},
}
```
Matched Drawings of Planar Graphs
E. Di Giacomo, W. Didimo, M.J. Kreveld, van, G. Liotta, and B. Speckmann.
Graph Drawing (15th International Symposium, GD'07, Sydney, Australia, September 23-26, 2007, Revised Papers), pp. 183—194, 2008.

－ Abstract

A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3-connected planar graphs or a 3-connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a planar graph of some special families—such as unlabeled level planar (ULP) graphs or the family of "carousel graphs"—are always matched drawable. Research partially supported by the MIUR Project "MAINSTREAM: Algorithms for massive information structures and data streams".
－ BibTeX
```
@article{matched-drawings-of-planar-graphs:2008,
    title = {Matched Drawings of Planar Graphs},
    author = {E. Di Giacomo and W. Didimo and M.J. Kreveld, van and G. Liotta and B. Speckmann},
    year = {2008},
    bookTitle = {Graph Drawing (15th International Symposium, GD'07, Sydney, Australia, September 23-26, 2007, Revised Papers)},
}
```
Polychromatic Colorings of Plane Graphs
N. Alon, R. Berke, K. Buchin, M. Buchin, P. Csorba, S. Shannigrahi, B. Speckmann, and P. Zumstein.
Proceedings 24th Annual ACM Symposium on Computational Geometry (SoCG'08, College Park MD, USA, June 9-11, 2008), pp. 338—345, 2008.

－ Abstract

We show that the vertices of any plane graph in which every face is of size at least g can be colored by (3g Àý 5)=4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than (3g+1)=4 colors. We further show that the problem of determining whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is NP-complete even for graphs in which all faces are of size 3 or 4 only. If all faces are of size 3 this can be decided in polynomial time.
－ BibTeX
```
@article{polychromatic-colorings-of-plane-graphs:2008,
    title = {Polychromatic Colorings of Plane Graphs},
    author = {N. Alon and R. Berke and K. Buchin and M. Buchin and P. Csorba and S. Shannigrahi and B. Speckmann and P. Zumstein},
    year = {2008},
    bookTitle = {Proceedings 24th Annual ACM Symposium on Computational Geometry (SoCG'08, College Park MD, USA, June 9-11, 2008)},
}
```
Triangulating and Guarding Realistic Polygons
G. Aloupis, P. Bose, V. Dujmovic, C.M. Gray, S. Langerman, and B. Speckmann.
Proceedings 20th Canadian Conference on Computational Geometry (CCCG'08, Montréal, Québec, August 13-15, 2008), pp. 107—110, 2008.

－ Abstract

We propose a new model of realistic input: k-guardable objects. An object is k-guardable if its boundary can be seen by k guards in the interior of the object. In this abstract, we describe a simple algorithm for triangulating k-guardable polygons. Our algorithm, which is easily implementable, takes linear time assuming that k is constant.
－ BibTeX
```
@article{triangulating-and-guarding-realistic-polygons:2008,
    title = {Triangulating and Guarding Realistic Polygons},
    author = {G. Aloupis and P. Bose and V. Dujmovic and C.M. Gray and S. Langerman and B. Speckmann},
    year = {2008},
    bookTitle = {Proceedings 20th Canadian Conference on Computational Geometry (CCCG'08, Montréal, Québec, August 13-15, 2008)},
}
```
Colour Patterns for Polychromatic Four-Colourings of Rectangular Subdivisions
H.J. Haverkort, M. Löffler, E. Mumford, M. O'Meara, J. Snoeyink, and B. Speckmann.
pp. 75—78, 2008.

－ Abstract

A non-degenerate rectangular subdivision is a subdivision of a rectangle into a set of non-overlapping rectangles S, such that no four rectangles meet in a point. We consider a problem that Katz and colleagues call strong polychromatic four-colouring: Colouring the vertices of the subdivision with four colours, such thateach rectangle of S has all colours among its four corners. By considering the possible colouring patterns, we can give short constructive proofs of colourabilityfor subdivisions that are sliceable or one-sided. We also present techniques and observations for nonsliceable, two-sided subdivisions.
－ BibTeX
```
@article{colour-patterns-for-polychromatic-four-colourings-of-rectangular-subdivisions:2008,
    title = {Colour Patterns for Polychromatic Four-Colourings of Rectangular Subdivisions},
    author = {H.J. Haverkort and M. Löffler and E. Mumford and M. O'Meara and J. Snoeyink and B. Speckmann},
    year = {2008},
    bookTitle = {},
}
```
[ PDF ]

2007

Algorithmic Aspects of Cartogram Computation
B. Speckmann.
Bulletin of the European Association for Theoretical Computer Science, EATCS, 92:33—43, 2007.

－ Abstract

－ BibTeX

@article{algorithmic-aspects-of-cartogram-computation:2007,
    title = {Algorithmic Aspects of Cartogram Computation},
    author = {B. Speckmann},
    year = {2007},
    bookTitle = {Bulletin of the European Association for Theoretical Computer Science, EATCS},
}

Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles
O. Aichholzer, C. Huemer, S. Kappes, B. Speckmann, and Cs.D. Tóth.
Graphs and Combinatorics, 23(5):481—507, 2007.

－ Abstract

We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties and give combinatorial bounds on their complexity. Our upper bounds depend on new Ramsey-type results concerning disjoint empty convex k-gons in point sets.
－ BibTeX
```
@article{decompositions-partitions-and-coverings-with-convex-polygons-and-pseudo-triangles:2007,
    title = {Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles},
    author = {O. Aichholzer and C. Huemer and S. Kappes and B. Speckmann and Cs.D. Tóth},
    year = {2007},
    bookTitle = {Graphs and Combinatorics},
}
```

Editorial (Special Issue on 21st European Workshop on Computational Geometry (EWCG 2005), Eindhoven, the Netherlands, March 9-11, 2005)
M. Berg, de, J. Gudmundsson, R. Oostrum, van, and B. Speckmann.
Computational Geometry, 36(1):1—, 2007.

－ Abstract

－ BibTeX

@article{editorial-special-issue-on-21st-european-workshop-on-computational-geometry-ewcg-2005-eindhoven-the-netherlands-march-9-11-2005-:2007,
    title = {Editorial (Special Issue on 21st European Workshop on Computational Geometry (EWCG 2005), Eindhoven, the Netherlands, March 9-11, 2005)},
    author = {M. Berg, de and J. Gudmundsson and R. Oostrum, van and B. Speckmann},
    year = {2007},
    bookTitle = {Computational Geometry},
}

Efficient Detection of Patterns in 2D Trajectories of Moving Points
J. Gudmundsson, M.J. Kreveld, van, and B. Speckmann.
GeoInformatica, 11(2):195—215, 2007.

－ Abstract

Moving point object data can be analyzed through the discovery of patterns in trajectories. We consider the computational efficiency of detecting four such spatio-temporal patterns, namely flock, leadership, convergence, and encounter, as defined by Laube et al., Finding REMO—detecting relative motion patterns in geospatial lifelines, 201–214, (2004). These patterns are large enough subgroups of the moving point objects that exhibit similar movement in the sense of direction, heading for the same location, and/or proximity. By the use of techniques from computational geometry, including approximation algorithms, we improve the running time bounds of existing algorithms to detect these patterns.
－ BibTeX
```
@article{efficient-detection-of-patterns-in-2d-trajectories-of-moving-points:2007,
    title = {Efficient Detection of Patterns in 2D Trajectories of Moving Points},
    author = {J. Gudmundsson and M.J. Kreveld, van and B. Speckmann},
    year = {2007},
    bookTitle = {GeoInformatica},
}
```
On Rectangular Cartograms
M.J. Kreveld, van and B. Speckmann.
Computational Geometry, 37(3):175—187, 2007.

－ Abstract

A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable (e.g., population). Good rectangular cartograms are hard to generate: The area specifications for each rectangle may make it impossible to realize correct adjacencies between the regions and so hamper the intuitive understanding of the map. We present the first algorithms for rectangular cartogram construction. Our algorithms depend on a precise formalization of region adjacencies and build upon existing VLSI layout algorithms. Furthermore, we characterize a non-trivial class of rectangular subdivisions for which exact cartograms can be computed efficiently. An implementation of our algorithms and various tests show that in practice, visually pleasing rectangular cartograms with small cartographic error can be generated effectively.
－ BibTeX
```
@article{on-rectangular-cartograms:2007,
    title = {On Rectangular Cartograms},
    author = {M.J. Kreveld, van and B. Speckmann},
    year = {2007},
    bookTitle = {Computational Geometry},
}
```
Edges and Switches, Tunnels and Bridges
D. Eppstein, M.J. Kreveld, van, E. Mumford, and B. Speckmann.
Proceedings of the 10th International Workshop on Algorithms and Data Structures (WADS 2007) 15-17 August 2007, Halifax, Nova Scotia, Canada, pp. 77—88, 2007.

－ Abstract

Edge casing is a well-known method to improve the readability of drawings of non-planar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than others. We formulate several optimization criteria that try to capture the concept of a "good" cased drawing. Further, we address the algorithmic question of how to turn a given drawing into an optimal cased drawing. For many of the resulting optimization problems, we either find polynomial time algorithms or NP-hardness results.
－ BibTeX
```
@article{edges-and-switches-tunnels-and-bridges:2007,
    title = {Edges and Switches, Tunnels and Bridges},
    author = {D. Eppstein and M.J. Kreveld, van and E. Mumford and B. Speckmann},
    year = {2007},
    bookTitle = {Proceedings of the 10th International Workshop on Algorithms and Data Structures (WADS 2007) 15-17 August 2007, Halifax, Nova Scotia, Canada},
}
```
[ PDF ]
Kinetic Kd-Trees and Longest-Side Kd-Trees
M.A. Abam, M. Berg, de, and B. Speckmann.
Proceedings of the 23rd Annual ACM Symposium on Computational Geometry (SoCG 2007) 6-8 June 2007, Geongju, South Korea, pp. 364—372, 2007.

－ Abstract

We propose a simple variant of kd-trees, called rank-based kd-trees, for sets of points in Rd. We show that a rank-based kd-tree, like an ordinary kd-tree, supports range search queries in O(n1-1/d + k) time, where k is the output size. The main advantage of rank-based kd-trees is that they can be efficiently kinetized: the KDS processes O(n2) events in the worst case, assuming that the points follow constant-degree algebraic trajectories, each event can be handled in O(log n) time, and each point is involved in O(1) certificates. We also propose a variant of longest-side kd-trees, called rank-based longest-side kd-trees (RBLS kd-trees, for short), for sets of points in R2. RBLS kd-trees can be kinetized efficiently as well and like longest-side kd-trees, RBLS kdtrees support nearest-neighbor, farthest-neighbor, and approximate range search queries in O((1/e) log2 n) time. The KDS processes O(n3 log n) events in the worst case, assuming that the points follow constant-degree algebraic trajectories; each event can be handled in O(log2 n) time, and each point is involved in O(log n) certificates.
－ BibTeX
```
@article{kinetic-kd-trees-and-longest-side-kd-trees:2007,
    title = {Kinetic Kd-Trees and Longest-Side Kd-Trees},
    author = {M.A. Abam and M. Berg, de and B. Speckmann},
    year = {2007},
    bookTitle = {Proceedings of the 23rd Annual ACM Symposium on Computational Geometry (SoCG 2007) 6-8 June 2007, Geongju, South Korea},
}
```
Maximizing Maximal Angles for Plane Straight-Line Graphs
O. Aichholzer, T. Hackl, M. Hoffmann, C. Huemer, A. Pór, F. Santos, B. Speckmann, and B. Vogtenhuber.
Proceedings of the 10th International Workshop on Algorithms and Data Structures (WADS 2007) 15-17 August 2007, Halifax, Nova Scotia, Canada, pp. 458—469, 2007.

－ Abstract

Let G¿=¿(S, E) be a plane straight-line graph on a finite point set in general position. The incident angles of a point p¿¿¿S in G are the angles between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight-line graph is called ¿-open if each vertex has an incident angle of size at least ¿. In this paper we study the following type of question: What is the maximum angle ¿ such that for any finite set of points in general position we can find a graph from a certain class of graphs on S that is ¿-open? In particular, we consider the classes of triangulations, spanning trees, and paths on S and give tight bounds in most cases.
－ BibTeX
```
@article{maximizing-maximal-angles-for-plane-straight-line-graphs:2007,
    title = {Maximizing Maximal Angles for Plane Straight-Line Graphs},
    author = {O. Aichholzer and T. Hackl and M. Hoffmann and C. Huemer and A. Pór and F. Santos and B. Speckmann and B. Vogtenhuber},
    year = {2007},
    bookTitle = {Proceedings of the 10th International Workshop on Algorithms and Data Structures (WADS 2007) 15-17 August 2007, Halifax, Nova Scotia, Canada},
}
```
[ PDF ]

On (Pointed) Minimum Weight Pseudo-Triangulations
O. Aichholzer, F. Aurenhammer, T. Hackl, and B. Speckmann.
Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG 2007) 20-22 August 2007, Ottawa, Canada, pp. 209—212, 2007.

－ Abstract

－ BibTeX

@article{on-pointed-minimum-weight-pseudo-triangulations:2007,
    title = {On (Pointed) Minimum Weight Pseudo-Triangulations},
    author = {O. Aichholzer and F. Aurenhammer and T. Hackl and B. Speckmann},
    year = {2007},
    bookTitle = {Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG 2007) 20-22 August 2007, Ottawa, Canada},
}

On the Number of Empty Pseudo-Triangles in Point Sets
M.J. Kreveld, van and B. Speckmann.
Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG 2007) 20-22 August 2007, Ottawa, Canada, pp. 37—40, 2007.

－ Abstract

We analyze the minimum and maximum number of empty pseudo-triangles defined by any planar point set. We consider the cases where the three convex vertices are fixed and where they are not fixed. Furthermore, the pseudo-triangles must either be star-shaped or can be arbitrary.
－ BibTeX
```
@article{on-the-number-of-empty-pseudo-triangles-in-point-sets:2007,
    title = {On the Number of Empty Pseudo-Triangles in Point Sets},
    author = {M.J. Kreveld, van and B. Speckmann},
    year = {2007},
    bookTitle = {Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG 2007) 20-22 August 2007, Ottawa, Canada},
}
```
Edges and Switches, Tunnels and Bridges
D. Eppstein, M.J. Kreveld, van, E. Mumford, and B. Speckmann.
pp. 146—149, 2007.

－ Abstract

Edge casing is a well-known method to improve the readability of drawings of non-planar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than others. We formulate several optimization criteria that try to capture the concept of a "good" cased drawing. Further, we address the algorithmic question of how to turn a given drawing into an optimal cased drawing. For many of the resulting optimization problems, we either find polynomial time algorithms or NP-hardness results.
－ BibTeX
```
@article{edges-and-switches-tunnels-and-bridges:2007,
    title = {Edges and Switches, Tunnels and Bridges},
    author = {D. Eppstein and M.J. Kreveld, van and E. Mumford and B. Speckmann},
    year = {2007},
    bookTitle = {},
}
```
[ PDF ]
Kinetic Kd-Trees
M.A. Abam, M. Berg, de, and B. Speckmann.
pp. 126—129, 2007.

－ Abstract

We propose a simple variant of kd-trees, called rankbased kd-trees, for sets of points in Rd. We show that a rank-based kd-tree, like an ordinary kd-tree, supports range search queries in O(n1-1/d + k) time, where k is the output size. The main advantage of rank-based kd-trees is that they can be efficiently kinetized: the KDS processes O(n2) events in the worst case, assuming that the points follow constantdegree algebraic trajectories, each event can be handled in O(log n) time, and each point is involved in O(1) certificates.
－ BibTeX
```
@article{kinetic-kd-trees:2007,
    title = {Kinetic Kd-Trees},
    author = {M.A. Abam and M. Berg, de and B. Speckmann},
    year = {2007},
    bookTitle = {},
}
```
[ PDF ]
Maximizing Maximal Angles for Plane Straight Line Graphs
O. Aichholzer, T. Hackl, M. Hoffmann, C. Huemer, F. Santos, B. Speckmann, and B. Vogtenhuber.
pp. 98—101, 2007.

－ Abstract

Let G = (S,E) be a plane straight line graph on a finite point set S R2 in general position. For a point p 2 S let the maximum incident angle of p in G be the maximum angle between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight line graph is called '-open if each vertex has an incident angle of size at least '. In this paper we study the following type of question: What is the maximum angle ' such that for any finite set S R2 of points in general position we can find a graph from a certain class of graphs on S that is '-open? In particular, we consider the classes of triangulations, spanning trees, and paths on S and give tight bounds in all but one cases.
－ BibTeX
```
@article{maximizing-maximal-angles-for-plane-straight-line-graphs:2007,
    title = {Maximizing Maximal Angles for Plane Straight Line Graphs},
    author = {O. Aichholzer and T. Hackl and M. Hoffmann and C. Huemer and F. Santos and B. Speckmann and B. Vogtenhuber},
    year = {2007},
    bookTitle = {},
}
```
[ PDF ]

2006

Area-Preserving Approximations of Polygonal Paths
P. Bose, S. Cabello, O. Cheong, J. Gudmundsson, M.J. Kreveld, van, and B. Speckmann.
Journal of Discrete Algorithms, 4(4):554—566, 2006.

－ Abstract

Let P be an x-monotone polygonal path in the plane. For a path Q that approximates P let WA(Q) be the area above P and below Q, and let WB(Q) be the area above Q and below P. Given P and an integer k, we show how to compute a path Q with at most k edges that minimizes WA(Q)+WB(Q). Given P and a cost C, we show how to find a path Q with the smallest possible number of edges such that WA(Q)+WB(Q)C. However, given P, an integer k, and a cost C, it is NP-hard to determine if a path Q with at most k edges exists such that max{WA(Q),WB(Q)}C. We describe an approximation algorithm for this setting. Finally, it is also NP-hard to decide whether a path Q exists such that |WA(Q)-WB(Q)|=0. Nevertheless, in this error measure we provide an algorithm for computing an optimal approximation up to an additive error.
－ BibTeX
```
@article{area-preserving-approximations-of-polygonal-paths:2006,
    title = {Area-Preserving Approximations of Polygonal Paths},
    author = {P. Bose and S. Cabello and O. Cheong and J. Gudmundsson and M.J. Kreveld, van and B. Speckmann},
    year = {2006},
    bookTitle = {Journal of Discrete Algorithms},
}
```
A Linear Programming Approach to Rectangular Cartograms
B. Speckmann, M.J. Kreveld, van, and S. Florisson.
Progress in Spatial Data Handling (Proceedings 12th International Symposium, SDH'06, Vienna, Austria, July 12-14, 2006), pp. 529—546, 2006.

－ Abstract

In [26], the first two authors of this paper presented the first algorithms to construct rectangular cartograms. The first step is to determine a representation of all regions by rectangles and the second—most important—step is to get the areas of all rectangles correct. This paper presents a new approach to the second step. It is based on alternatingly solving linear programs on the x-coordinates and the y-coordinates of the sides of the rectangles. Our algorithm gives cartograms with considerably lower error and better visual qualities than previous approaches. It also handles countries that cannot be present in any purely rectangular cartogram and it introduces a new way of controlling incorrect adjacencies of countries. Our implementation computes aesthetically pleasing rectangular and nearly rectangular cartograms, for instance depicting the 152 countries of the World that have population over one million.
－ BibTeX
```
@article{a-linear-programming-approach-to-rectangular-cartograms:2006,
    title = {A Linear Programming Approach to Rectangular Cartograms},
    author = {B. Speckmann and M.J. Kreveld, van and S. Florisson},
    year = {2006},
    bookTitle = {Progress in Spatial Data Handling (Proceedings 12th International Symposium, SDH'06, Vienna, Austria, July 12-14, 2006)},
}
```
Algorithmic Aspects of Proportional Symbol Maps
S. Cabello, H.J. Haverkort, M.J. Kreveld, van, and B. Speckmann.
Algorithms - ESA 2006 (Proceedings 14th Annual European Symposium, Zürich, Switzerland, September 11-13, 2006), pp. 720—731, 2006.

－ Abstract

Proportional symbol maps visualize numerical data associated with point locations by placing a scaled symbol—typically opaque disks or squares—at the corresponding point on a map. Overlapping symbols need to be drawn in such a way that the user can still judge their relative sizes accurately. We identify two types of suitable drawings: physically realizable drawings and stacking drawings. For these we study the following two problems: Max-Min—maximize the minimum visible boundary length of each symbol—and Max-Total—maximize the total visible boundary length over all symbols. We show that both problems are NP-hard for physically realizable drawings. Max-Min can be solved in O(n2 log n) time for stacking drawings, which can be improved to O(n log n) or O(n log2 n) time when the input has certain properties. We also experimented with four methods to compute stacking drawings: our solution to the Max-Min problem performs best on the data sets considered.
－ BibTeX
```
@article{algorithmic-aspects-of-proportional-symbol-maps:2006,
    title = {Algorithmic Aspects of Proportional Symbol Maps},
    author = {S. Cabello and H.J. Haverkort and M.J. Kreveld, van and B. Speckmann},
    year = {2006},
    bookTitle = {Algorithms - ESA 2006 (Proceedings 14th Annual European Symposium, Zürich, Switzerland, September 11-13, 2006)},
}
```
Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles
O. Aichholzer, C. Huemer, S. Kappes, B. Speckmann, and Cs.D. Tóth.
Mathematical Foundations of Computer Science (Proceedings 31st International Symposium, MFCS 2006, Stará Lesná, Slovakia, August 28-September 1, 2006), pp. 86—97, 2006.

－ Abstract

We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings.We establish some basic properties and give combinatorial bounds on their complexity. Our upper bounds depend on new Ramsey-type results concerning disjoint empty convex k-gons in point sets.
－ BibTeX
```
@article{decompositions-partitions-and-coverings-with-convex-polygons-and-pseudo-triangles:2006,
    title = {Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles},
    author = {O. Aichholzer and C. Huemer and S. Kappes and B. Speckmann and Cs.D. Tóth},
    year = {2006},
    bookTitle = {Mathematical Foundations of Computer Science (Proceedings 31st International Symposium, MFCS 2006, Stará Lesná, Slovakia, August 28-September 1, 2006)},
}
```
Kinetic Collision Detection for Convex Fat Objects
M.A. Abam, M. Berg, de, S.H. Poon, and B. Speckmann.
Algorithms - ESA 2006 (Proceedings 14th Annual European Symposium, Zürich, Switzerland, September 11-13, 2006), pp. 4—15, 2006.

－ Abstract

We design compact and responsive kinetic data structures for detecting collisions between n convex fat objects in 3-dimensional space that can have arbitrary sizes. Our main results are: (i) If the objects are 3-dimensional balls that roll on a plane, then we can detect collisions with a KDS of size O(n log n) that can handle events in O(log n) time. This structure processes O(n2) events in the worst case, assuming that the objects follow constant-degree algebraic trajectories. (ii) If the objects are convex fat 3-dimensional objects of constant complexity that are free-flying in R3, then we can detect collisions with a KDS of O(n log6 n) size that can handle events in O(log6 n) time. This structure processes O(n2) events in the worst case, assuming that the objects follow constant-degree algebraic trajectories. If the objects have similar sizes then the size of the KDS becomes O(n) and events can be handled in O(1) time.
－ BibTeX
```
@article{kinetic-collision-detection-for-convex-fat-objects:2006,
    title = {Kinetic Collision Detection for Convex Fat Objects},
    author = {M.A. Abam and M. Berg, de and S.H. Poon and B. Speckmann},
    year = {2006},
    bookTitle = {Algorithms - ESA 2006 (Proceedings 14th Annual European Symposium, Zürich, Switzerland, September 11-13, 2006)},
}
```

On Rectilinear Duals for Vertex-Weighted Plane Graphs
M. Berg, de, E. Mumford, and B. Speckmann.
Graph Drawing (13th International Symposium, GD'05, Limerick, Ireland, September 12-14, 2005, Revised papers), pp. 61—72, 2006.

－ BibTeX

@article{on-rectilinear-duals-for-vertex-weighted-plane-graphs:2006,
    title = {On Rectilinear Duals for Vertex-Weighted Plane Graphs},
    author = {M. Berg, de and E. Mumford and B. Speckmann},
    year = {2006},
    bookTitle = {Graph Drawing (13th International Symposium, GD'05, Limerick, Ireland, September 12-14, 2005, Revised papers)},
}

Optimal BSPs and Rectilinear Cartograms
M. Berg, de, E. Mumford, and B. Speckmann.
Proceedings 14th International Symposium on Advances in Geographic Information Systems (ACM-GIS'06, Arlington VA, USA, November 10-11, 2006), pp. 19—26, 2006.

－ Abstract

A cartogram is a thematic map that visualizes statistical data about a set of regions like countries, states or provinces. The size of a region in a cartogram corresponds to a particular geographic variable, for example, population. We present an algorithm for constructing rectilinear cartograms (each region is represented by a rectilinear polygon) with zero cartographic error and correct region adjacencies, and we test our algorithm on various data sets. It produces regions of very small complexity---in fact, most regions are rectangles---while still ensuring both exact areas and correct adjacencies for all regions.Our algorithm uses a novel subroutine that is interesting in its own right, namely a polynomial-time algorithm for computing optimal binary space partitions (BSPs) for rectilinear maps. This algorithm works for a general class of optimality criteria, including size and depth. We use this generality in our application to computing cartograms, where we apply a dedicated cost function leading to BSP's amenable to the constructing of high-quality cartograms.
－ BibTeX
```
@article{optimal-bsps-and-rectilinear-cartograms:2006,
    title = {Optimal BSPs and Rectilinear Cartograms},
    author = {M. Berg, de and E. Mumford and B. Speckmann},
    year = {2006},
    bookTitle = {Proceedings 14th International Symposium on Advances in Geographic Information Systems (ACM-GIS'06, Arlington VA, USA, November 10-11, 2006)},
}
```
Kinetic Collision Detection for Balls Rolling on a Plane
M.A. Abam, M. Berg, de, S.H. Poon, and B. Speckmann.
pp. 55—58, 2006.

－ Abstract

This abstract presents a first step towards kinetic col- lision detection in 3 dimensions. In particular, we design a compact and responsive kinetic data struc- ture (KDS) for detecting collisions between n balls of arbitrary sizes rolling on a plane. The KDS has size O(n log n) and can handle events in O(log n) time. The structure processes O(n2) events in the worst case, assuming that the objects follow low-degree al- gebraic trajectories. The full paper [1] presents ad- ditional results for convex fat 3-dimensional objects that are free-flying in R3.
－ BibTeX
```
@article{kinetic-collision-detection-for-balls-rolling-on-a-plane:2006,
    title = {Kinetic Collision Detection for Balls Rolling on a Plane},
    author = {M.A. Abam and M. Berg, de and S.H. Poon and B. Speckmann},
    year = {2006},
    bookTitle = {},
}
```
[ PDF ]

On the Number of Pseudo-Triangulations of Certain Point Sets
O. Aichholzer, D. Orden, F. Santos, and B. Speckmann.
2006.

－ BibTeX

@article{on-the-number-of-pseudo-triangulations-of-certain-point-sets:2006,
    title = {On the Number of Pseudo-Triangulations of Certain Point Sets},
    author = {O. Aichholzer and D. Orden and F. Santos and B. Speckmann},
    year = {2006},
    bookTitle = {},
}

2005

Computational Geometry: Fundamental Structures
M. Berg, de and B. Speckmann.
Handbook of Data Structures and Applications, pp. 62—1—62—20, 2005.

－ Abstract

－ BibTeX

@article{computational-geometry-fundamental-structures:2005,
    title = {Computational Geometry: Fundamental Structures},
    author = {M. Berg, de and B. Speckmann},
    year = {2005},
    bookTitle = {Handbook of Data Structures and Applications},
}

Allocating Vertex $\pi$-Guards in Simple Polygons Via Pseudo-Triangulations
B. Speckmann and Cs.D. Tóth.
Discrete and Computational Geometry, 33(2):345—364, 2005.

－ BibTeX

@article{allocating-vertex-pi-guards-in-simple-polygons-via-pseudo-triangulations:2005,
    title = {Allocating Vertex $\pi$-Guards in Simple Polygons Via Pseudo-Triangulations},
    author = {B. Speckmann and Cs.D. Tóth},
    year = {2005},
    bookTitle = {Discrete and Computational Geometry},
}

Off-Line Admission Control for Advance Reservations in Star Networks
U. Adamy, T. Erlebach, D. Mitsche, I. Schurr, B. Speckmann, and E. Welzl.
Approximation and Online Algorithms (2nd International Workshop, WAOA 2004, Bergen, Norway, September 14-16, 2004, Revised Selected Papers), pp. 211—224, 2005.

－ Abstract

Given a network together with a set of connection requests, call admission control is the problem of deciding which calls to accept and which ones to reject in order to maximize the total profit of the accepted requests. We consider call admission control problems with advance reservations in star networks. For the most general variant we present a constant-factor approximation algorithm resolving an open problem due to Erlebach. Our method is randomized and achieves an approximation ratio of 1/18. It can be generalized to accommodate call alternatives, in which case the approximation ratio is 1/24. We show how our method can be derandomized. In addition we prove that call admission control in star networks is -hard even for very restricted variants of the problem.
－ BibTeX
```
@article{off-line-admission-control-for-advance-reservations-in-star-networks:2005,
    title = {Off-Line Admission Control for Advance Reservations in Star Networks},
    author = {U. Adamy and T. Erlebach and D. Mitsche and I. Schurr and B. Speckmann and E. Welzl},
    year = {2005},
    bookTitle = {Approximation and Online Algorithms (2nd International Workshop, WAOA 2004, Bergen, Norway, September 14-16, 2004, Revised Selected Papers)},
}
```

Rectangular Cartogram Computation with Sea Regions
M.J. Kreveld, van and B. Speckmann.
Proceedings 22nd International Cartographic Conference (XXII ICC, A Coruña, Spain, July 9-16, 2005), CD-ROM, 2005.

－ BibTeX

@article{rectangular-cartogram-computation-with-sea-regions:2005,
    title = {Rectangular Cartogram Computation with Sea Regions},
    author = {M.J. Kreveld, van and B. Speckmann},
    year = {2005},
    bookTitle = {Proceedings 22nd International Cartographic Conference (XXII ICC, A Coruña, Spain, July 9-16, 2005), CD-ROM},
}

Rectangular Cartograms: Construction and Animation
S. Florisson, M.J. Kreveld, van, and B. Speckmann.
Proceedings 14th Annual Multimedia Review of Computational Geometry (part of 21st Annual ACM Symposium on Computational Geometry, Pisa, Italy, June 6-8, 2005), pp. 372—373, 2005.

－ BibTeX

@article{rectangular-cartograms-construction-and-animation:2005,
    title = {Rectangular Cartograms: Construction and Animation},
    author = {S. Florisson and M.J. Kreveld, van and B. Speckmann},
    year = {2005},
    bookTitle = {Proceedings 14th Annual Multimedia Review of Computational Geometry (part of 21st Annual ACM Symposium on Computational Geometry, Pisa, Italy, June 6-8, 2005)},
}

On Pseudo-Convex Decompositions, Partitions, and Coverings
O. Aichholzer, C. Huemer, S. Renkl, B. Speckmann, and Cs.D. Tóth.
pp. 89—92, 2005.

－ Abstract

We introduce pseudo-convex decompositions, partitions, and coverings for planar point sets. They are natural extensions of their convex counterparts and use both convex polygons and pseudo-triangles. We discuss some of their basic combinatorial properties and establish upper and lower bounds on their complexity.
－ BibTeX
```
@article{on-pseudo-convex-decompositions-partitions-and-coverings:2005,
    title = {On Pseudo-Convex Decompositions, Partitions, and Coverings},
    author = {O. Aichholzer and C. Huemer and S. Renkl and B. Speckmann and Cs.D. Tóth},
    year = {2005},
    bookTitle = {},
}
```
[ PDF ]

2004

Convexity Minimizes Pseudo-Triangulations
O. Aichholzer, F. Aurenhammer, H. Krasser, and B. Speckmann.
Computational Geometry, 28(1):3—10, 2004.

－ BibTeX

@article{convexity-minimizes-pseudo-triangulations:2004,
    title = {Convexity Minimizes Pseudo-Triangulations},
    author = {O. Aichholzer and F. Aurenhammer and H. Krasser and B. Speckmann},
    year = {2004},
    bookTitle = {Computational Geometry},
}

Bounds on the k-Neighborhood for Locally Uniform Sampled Surfaces
M. Andersson, J. Giesen, M. Pauly, and B. Speckmann.
Proceedings of Symposium on Point-Based Graphics (SPBG 2004, Zurich, Switzerland, June 2-4, 2004), pp. 167—171, 233, 2004.

－ Abstract

Given a locally uniform sample set P of a smooth surface S. We derive upper and lower bounds on the number k of nearest neighbors of a sample point p that have to be chosen from P such that this neighborhood contains all restricted Delaunay neighbors of p. In contrast to the trivial lower bound, the upper bound indicates that a sampling condition that is used in many computational geometry proofs is quite reasonable from a practical point of view.
－ BibTeX
```
@article{bounds-on-the-k-neighborhood-for-locally-uniform-sampled-surfaces:2004,
    title = {Bounds on the k-Neighborhood for Locally Uniform Sampled Surfaces},
    author = {M. Andersson and J. Giesen and M. Pauly and B. Speckmann},
    year = {2004},
    bookTitle = {Proceedings of Symposium on Point-Based Graphics (SPBG 2004, Zurich, Switzerland, June 2-4, 2004)},
}
```
[ PDF ]

Efficient Detection of Motion Patterns in Spatio-Temporal Data Sets
J. Gudmundsson, M.J. Kreveld, van, and B. Speckmann.
Proceedings 12th International Workshop on Geographic Information Systems (ACM-GIS 2004, Washington DC, November 12-13, 2004), pp. 250—257, 2004.

－ BibTeX

@article{efficient-detection-of-motion-patterns-in-spatio-temporal-data-sets:2004,
    title = {Efficient Detection of Motion Patterns in Spatio-Temporal Data Sets},
    author = {J. Gudmundsson and M.J. Kreveld, van and B. Speckmann},
    year = {2004},
    bookTitle = {Proceedings 12th International Workshop on Geographic Information Systems (ACM-GIS 2004, Washington DC, November 12-13, 2004)},
}

On Rectangular Cartograms
M.J. van Kreveld and B. Speckmann.
Proceedings of the Tenth Annual Conference of the Advanced School for Computing and Imaging (ASCI 2004), Port Zélande, Ouddorp, The Netherlands, June 2-4, 2004, pp. 11—16, 2004.

－ BibTeX

@article{on-rectangular-cartograms:2004,
    title = {On Rectangular Cartograms},
    author = {M.J. van Kreveld and B. Speckmann},
    year = {2004},
    bookTitle = {Proceedings of the Tenth Annual Conference of the Advanced School for Computing and Imaging (ASCI 2004), Port Zélande, Ouddorp, The Netherlands, June 2-4, 2004},
}

On Rectangular Cartograms
M.J. Kreveld, van and B. Speckmann.
Algorithms - ESA 2004 : proceedings 12th annual european symposium, Bergen, Norway, September 14-17, 2004, pp. 724—735, 2004.

－ Abstract

A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable (e.g., population). Good rectangular cartograms are hard to generate: The area specifications for each rectangle may make it impossible to realize correct adjacencies between the regions and so hamper the intuitive understanding of the map. Here we present the first algorithms for rectangular cartogram construction. Our algorithms depend on a precise formalization of region adjacencies and are building upon existing VLSI layout algorithms. Furthermore, we characterize a non-trivial class of rectangular subdivisions for which exact cartograms can be efficiently computed. An implementation of our algorithms and various tests show that in practice, visually pleasing rectangular cartograms with small cartographic error can be effectively generated.
－ BibTeX
```
@article{on-rectangular-cartograms:2004,
    title = {On Rectangular Cartograms},
    author = {M.J. Kreveld, van and B. Speckmann},
    year = {2004},
    bookTitle = {Algorithms - ESA 2004 : proceedings 12th annual european symposium, Bergen, Norway, September 14-17, 2004},
}
```
Pointed Binary Encompassing Trees
M. Hoffmann, B. Speckmann, and Cs.D. Tóth.
Algorithm Theory - SWAT 2004 (Proceedings 9th Scandinavian Workshop on Algorithm Theory, Humlebaek, Denmark, July 8-10, 2004), pp. 442—454, 2004.

－ Abstract

We show that for any set of disjoint line segments in the plane there exists a pointed binary encompassing tree T, that is, a spanning tree on the segment endpoints that contains all input segments, has maximum degree three, and every vertex v $\in$ T is pointed, that is, v has an incident angle greater than $\pi$. Such a tree can be completed to a minimum pseudo-triangulation. In particular, it follows that every set of disjoint line segments has a minimum pseudo-triangulation of bounded vertex degree.
－ BibTeX
```
@article{pointed-binary-encompassing-trees:2004,
    title = {Pointed Binary Encompassing Trees},
    author = {M. Hoffmann and B. Speckmann and Cs.D. Tóth},
    year = {2004},
    bookTitle = {Algorithm Theory - SWAT 2004 (Proceedings 9th Scandinavian Workshop on Algorithm Theory, Humlebaek, Denmark, July 8-10, 2004)},
}
```

On the Number of Pseudo-Triangulations of Certain Point Sets
O. Aichholzer, D. Orden, F. Santos, and B. Speckmann.
pp. 119—122, 2004.

－ BibTeX

@article{on-the-number-of-pseudo-triangulations-of-certain-point-sets:2004,
    title = {On the Number of Pseudo-Triangulations of Certain Point Sets},
    author = {O. Aichholzer and D. Orden and F. Santos and B. Speckmann},
    year = {2004},
    bookTitle = {},
}

2003

Tight Degree Bounds for Pseudo-Triangulations of Points
L. Kettner, D. Kirkpatrick, A. Mantler, J. Snoeyink, B. Speckmann, and F. Takeuchi.
Computational Geometry, 25(1-2):3—12, 2003.

－ BibTeX

@article{tight-degree-bounds-for-pseudo-triangulations-of-points:2003,
    title = {Tight Degree Bounds for Pseudo-Triangulations of Points},
    author = {L. Kettner and D. Kirkpatrick and A. Mantler and J. Snoeyink and B. Speckmann and F. Takeuchi},
    year = {2003},
    bookTitle = {Computational Geometry},
}

Allocating Vertex $\pi$-Guards in Simple Polygons Via Pseudo-Triangulations
B. Speckmann and Cs.D. Tóth.
Proceedings 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2003, Baltimore MD, USA, January 12-14, 2003), pp. 109—118, 2003.

－ BibTeX

@article{allocating-vertex-pi-guards-in-simple-polygons-via-pseudo-triangulations:2003,
    title = {Allocating Vertex $\pi$-Guards in Simple Polygons Via Pseudo-Triangulations},
    author = {B. Speckmann and Cs.D. Tóth},
    year = {2003},
    bookTitle = {Proceedings 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2003, Baltimore MD, USA, January 12-14, 2003)},
}

Degree Bounds for Constrained Pseudo-Triangulations
O. Aichholzer, M. Hoffmann, B. Speckmann, and Cs.D. Tóth.
Proceedings of the 15th Canadian Conference on Computational Geometry (CCCG'03, Halifax, Nova Scotia, Canada, August 11-13, 2003), Electronic proceedings, pp. 155—158, 2003.

－ Abstract

We introduce the concept of a constrained pointed pseudo-triangulation TG of a point set S with respect to a pointed planar straight line graph G = (S,E). For the case that G forms a simple polygon P with vertex set S we give tight bounds on the vertex degree of TG.
－ BibTeX
```
@article{degree-bounds-for-constrained-pseudo-triangulations:2003,
    title = {Degree Bounds for Constrained Pseudo-Triangulations},
    author = {O. Aichholzer and M. Hoffmann and B. Speckmann and Cs.D. Tóth},
    year = {2003},
    bookTitle = {Proceedings of the 15th Canadian Conference on Computational Geometry (CCCG'03, Halifax, Nova Scotia, Canada, August 11-13, 2003), Electronic proceedings},
}
```
On the Number of Pseudo-Triangulations of Certain Point Sets
O. Aichholzer, D. Orden, F. Santos, and B. Speckmann.
Proc. 15th Canadian Conference on Computational Geometry (CCCG), pp. 141—144, 2003.

－ Abstract

We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it in two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically 12nn£(1) pointed pseudo-triangulations, which lies signi¯cantly above the maximum number of triangulations in a planar point set known so far.
－ BibTeX
```
@article{on-the-number-of-pseudo-triangulations-of-certain-point-sets:2003,
    title = {On the Number of Pseudo-Triangulations of Certain Point Sets},
    author = {O. Aichholzer and D. Orden and F. Santos and B. Speckmann},
    year = {2003},
    bookTitle = {Proc. 15th Canadian Conference on Computational Geometry (CCCG)},
}
```
[ PDF ]
The Zigzag Path of a Pseudo-Triangulation
O. Aichholzer, G. Rote, B. Speckmann, and I. Streinu.
Algorithms and Data Structures (Proceedings 8th International Workshop, WADS 2003, Ottawa, Ontario, Canada, July 30-August 1, 2003), pp. 377—388, 2003.

－ Abstract

We define the zigzag path of a pseudo-triangulation, a concept generalizing the path of a triangulation of a point set. The pseudo-triangulation zigzag path allows us to use divide-and-conquer type of approaches for suitable (i.e., decomposable) problems on pseudo-triangulations. For this we provide an algorithm that enumerates all pseudo-triangulation zigzag paths (of all pseudo-triangulations of a given point set with respect to a given line) in O(n 2) time per path and O(n 2) space, where n is the number of points. We illustrate applications of our scheme which include a novel algorithm to count the number of pseudo-triangulations of a point set.
－ BibTeX
```
@article{the-zigzag-path-of-a-pseudo-triangulation:2003,
    title = {The Zigzag Path of a Pseudo-Triangulation},
    author = {O. Aichholzer and G. Rote and B. Speckmann and I. Streinu},
    year = {2003},
    bookTitle = {Algorithms and Data Structures (Proceedings 8th International Workshop, WADS 2003, Ottawa, Ontario, Canada, July 30-August 1, 2003)},
}
```

2002

Cutting a Country for Smallest Square Fit
Marc van Kreveld and Bettina Speckmann.
Algorithms and Computation - 13th International Symposium, ISAAC 2002, Proceedings, pp. 91—102, 2002.

－ Abstract

We study the problem of cutting a simple polygon with n vertices into two pieces such that - if we reposition one piece disjoint of the other, without rotation - they have the minimum possible bounding square. If we cut with a single horizontal or vertical segment, then we can compute an optimal solution for a convex polygon with n vertices in O(n) time. For simple polygons we give an O(n4α(n) logn) time algorithm.
－ BibTeX
```
@article{cutting-a-country-for-smallest-square-fit:2002,
 title = {Cutting a Country for Smallest Square Fit},
 author = {Marc van Kreveld and Bettina Speckmann},
 year = {2002},
 bookTitle = {Algorithms and Computation - 13th International Symposium, ISAAC 2002, Proceedings},
}
```

2000

Kinetic Collision Detection for Simple Polygons
D. Kirkpatrick, J. Snoeyink, and B. Speckmann.
Proceedings 16th Annual ACM Symposium on Computational Geometry (Clear Water Bay, Hong Kong, June 12-14, 2000), pp. 322—330, 2000.

－ BibTeX

@article{kinetic-collision-detection-for-simple-polygons:2000,
    title = {Kinetic Collision Detection for Simple Polygons},
    author = {D. Kirkpatrick and J. Snoeyink and B. Speckmann},
    year = {2000},
    bookTitle = {Proceedings 16th Annual ACM Symposium on Computational Geometry (Clear Water Bay, Hong Kong, June 12-14, 2000)},
}