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**Published in:** LIPIcs, Volume 208, 11th International Conference on Geographic Information Science (GIScience 2021) - Part II

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.

Bram Custers, Wouter Meulemans, Bettina Speckmann, and Kevin Verbeek. Coordinated Schematization for Visualizing Mobility Patterns on Networks. In 11th International Conference on Geographic Information Science (GIScience 2021) - Part II. Leibniz International Proceedings in Informatics (LIPIcs), Volume 208, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{custers_et_al:LIPIcs.GIScience.2021.II.7, author = {Custers, Bram and Meulemans, Wouter and Speckmann, Bettina and Verbeek, Kevin}, title = {{Coordinated Schematization for Visualizing Mobility Patterns on Networks}}, booktitle = {11th International Conference on Geographic Information Science (GIScience 2021) - Part II}, pages = {7:1--7:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-208-2}, ISSN = {1868-8969}, year = {2021}, volume = {208}, editor = {Janowicz, Krzysztof and Verstegen, Judith A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GIScience.2021.II.7}, URN = {urn:nbn:de:0030-drops-147665}, doi = {10.4230/LIPIcs.GIScience.2021.II.7}, annote = {Keywords: Trajectories, Visualization, Schematization} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

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.

Pantea Haghighatkhah, Wouter Meulemans, Bettina Speckmann, Jérôme Urhausen, and Kevin Verbeek. Obstructing Classification via Projection. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 56:1-56:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{haghighatkhah_et_al:LIPIcs.MFCS.2021.56, author = {Haghighatkhah, Pantea and Meulemans, Wouter and Speckmann, Bettina and Urhausen, J\'{e}r\^{o}me and Verbeek, Kevin}, title = {{Obstructing Classification via Projection}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {56:1--56:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.56}, URN = {urn:nbn:de:0030-drops-144965}, doi = {10.4230/LIPIcs.MFCS.2021.56}, annote = {Keywords: Projection, classification, models of learning} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

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.

Tim Ophelders, Ignaz Rutter, Bettina Speckmann, and Kevin Verbeek. Polygon-Universal Graphs. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ophelders_et_al:LIPIcs.SoCG.2021.55, author = {Ophelders, Tim and Rutter, Ignaz and Speckmann, Bettina and Verbeek, Kevin}, title = {{Polygon-Universal Graphs}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {55:1--55:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.55}, URN = {urn:nbn:de:0030-drops-138540}, doi = {10.4230/LIPIcs.SoCG.2021.55}, annote = {Keywords: Graph drawing, partial drawing extension, simple polygon} }

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**Published in:** LIPIcs, Volume 177, 11th International Conference on Geographic Information Science (GIScience 2021) - Part I (2020)

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.

Marc van Kreveld, Tim Ophelders, Willem Sonke, Bettina Speckmann, and Kevin Verbeek. Volume from Outlines on Terrains. In 11th International Conference on Geographic Information Science (GIScience 2021) - Part I. Leibniz International Proceedings in Informatics (LIPIcs), Volume 177, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{vankreveld_et_al:LIPIcs.GIScience.2021.I.16, author = {van Kreveld, Marc and Ophelders, Tim and Sonke, Willem and Speckmann, Bettina and Verbeek, Kevin}, title = {{Volume from Outlines on Terrains}}, booktitle = {11th International Conference on Geographic Information Science (GIScience 2021) - Part I}, pages = {16:1--16:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-166-5}, ISSN = {1868-8969}, year = {2020}, volume = {177}, editor = {Janowicz, Krzysztof and Verstegen, Judith A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GIScience.2021.I.16}, URN = {urn:nbn:de:0030-drops-130512}, doi = {10.4230/LIPIcs.GIScience.2021.I.16}, annote = {Keywords: Terrain model, similarity, volume, computation} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We describe an algorithm that morphs between two planar orthogonal drawings Gamma_I and Gamma_O of a connected graph G, while preserving planarity and orthogonality. Necessarily Gamma_I and Gamma_O share the same combinatorial embedding. 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. [Biedl et al., 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 Gamma_O. We can find corresponding wires in Gamma_I that share topological properties with the wires in Gamma_O. The structural difference between the two drawings can be captured by the spirality of the wires in Gamma_I, which guides our morph from Gamma_I to Gamma_O.

Arthur van Goethem and Kevin Verbeek. Optimal Morphs of Planar Orthogonal Drawings. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{vangoethem_et_al:LIPIcs.SoCG.2018.42, author = {van Goethem, Arthur and Verbeek, Kevin}, title = {{Optimal Morphs of Planar Orthogonal Drawings}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {42:1--42:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.42}, URN = {urn:nbn:de:0030-drops-87550}, doi = {10.4230/LIPIcs.SoCG.2018.42}, annote = {Keywords: Homotopy, Morphing, Orthogonal drawing, Spirality} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

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.

Irina Kostitsyna, Bettina Speckmann, and Kevin Verbeek. Non-Crossing Geometric Steiner Arborescences. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kostitsyna_et_al:LIPIcs.ISAAC.2017.54, author = {Kostitsyna, Irina and Speckmann, Bettina and Verbeek, Kevin}, title = {{Non-Crossing Geometric Steiner Arborescences}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {54:1--54:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.54}, URN = {urn:nbn:de:0030-drops-82342}, doi = {10.4230/LIPIcs.ISAAC.2017.54}, annote = {Keywords: Steiner arborescences, non-crossing drawing, rectilinear, angle-restricted} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

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 delta-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 delta-different network that respects this order in polynomial time. We implemented our approach and applied it to simulated and real-world braided rivers.

Maarten Kleinhans, Marc van Kreveld, Tim Ophelders, Willem Sonke, Bettina Speckmann, and Kevin Verbeek. Computing Representative Networks for Braided Rivers. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kleinhans_et_al:LIPIcs.SoCG.2017.48, author = {Kleinhans, Maarten and van Kreveld, Marc and Ophelders, Tim and Sonke, Willem and Speckmann, Bettina and Verbeek, Kevin}, title = {{Computing Representative Networks for Braided Rivers}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {48:1--48:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.48}, URN = {urn:nbn:de:0030-drops-72204}, doi = {10.4230/LIPIcs.SoCG.2017.48}, annote = {Keywords: braided rivers, Morse-Smale complex, persistence, network extraction, polyhedral terrain} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

The Most Likely Voronoi Diagram is a generalization of the well known Voronoi Diagrams to a stochastic setting, where a stochastic point is a point associated with a given probability of existence, and the cell for such a point is the set of points which would classify the given point as its most likely nearest neighbor. We investigate the complexity of this subdivision of space in d dimensions. We show that in the general case, the complexity of such a subdivision is Omega(n^{2d}) where n is the number of points. This settles an open question raised in a recent (ISAAC 2014) paper of Suri and Verbeek, which first defined the Most Likely Voronoi Diagram. We also show that when the probabilities are assigned using a random permutation of a fixed set of values, in expectation the complexity is only ~O(n^{ceil{d/2}}) where the ~O(*) means that logarithmic factors are suppressed. In the worst case, this bound is tight up to polylog factors.

Nirman Kumar, Benjamin Raichel, Subhash Suri, and Kevin Verbeek. Most Likely Voronoi Diagrams in Higher Dimensions. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kumar_et_al:LIPIcs.FSTTCS.2016.31, author = {Kumar, Nirman and Raichel, Benjamin and Suri, Subhash and Verbeek, Kevin}, title = {{Most Likely Voronoi Diagrams in Higher Dimensions}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.31}, URN = {urn:nbn:de:0030-drops-68667}, doi = {10.4230/LIPIcs.FSTTCS.2016.31}, annote = {Keywords: Uncertainty, Lower bounds, Voronoi Diagrams, Stochastic} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

We show how to represent a simple polygon P by a (pixel-based) grid polygon Q that is simple and whose Hausdorff or Fréchet distance to P is small. For any simple polygon P, a grid polygon exists with constant Hausdorff distance between their boundaries and their interiors. Moreover, we show that with a realistic input assumption we can also realize constant Fréchet distance between the boundaries. We present algorithms accompanying these constructions, heuristics to improve their output while keeping the distance bounds, and experiments to assess the output.

Quirijn W. Bouts, Irina Irina Kostitsyna, Marc van Kreveld, Wouter Meulemans, Willem Sonke, and Kevin Verbeek. Mapping Polygons to the Grid with Small Hausdorff and Fréchet Distance. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bouts_et_al:LIPIcs.ESA.2016.22, author = {Bouts, Quirijn W. and Irina Kostitsyna, Irina and van Kreveld, Marc and Meulemans, Wouter and Sonke, Willem and Verbeek, Kevin}, title = {{Mapping Polygons to the Grid with Small Hausdorff and Fr\'{e}chet Distance}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {22:1--22:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.22}, URN = {urn:nbn:de:0030-drops-63738}, doi = {10.4230/LIPIcs.ESA.2016.22}, annote = {Keywords: grid mapping, Hausdorff distance, Fr\'{e}chet distance, digital geometry} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

The chromatic art gallery problem asks for the minimum number of "colors" t so that a collection of point guards, each assigned one of the t colors, can see the entire polygon subject to some conditions on the colors visible to each point. In this paper, we explore this problem for orthogonal polygons using orthogonal visibility - two points p and q are mutually visible if the smallest axis-aligned rectangle containing them lies within the polygon. Our main result establishes that for a conflict-free guarding of an orthogonal n-gon, in which at least one of the colors seen by every point is unique, the number of colors is Theta(loglog n). By contrast, the best upper bound for orthogonal polygons under standard (non-orthogonal) visibility is O(log n) colors. We also show that the number of colors needed for strong guarding of simple orthogonal polygons, where all the colors visible to a point are unique, is Theta(log n). Finally, our techniques also help us establish the first non-trivial lower bound of Omega(loglog n / logloglog n) for conflict-free guarding under standard visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau.

Frank Hoffmann, Klaus Kriegel, Subhash Suri, Kevin Verbeek, and Max Willert. Tight Bounds for Conflict-Free Chromatic Guarding of Orthogonal Art Galleries. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 421-435, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hoffmann_et_al:LIPIcs.SOCG.2015.421, author = {Hoffmann, Frank and Kriegel, Klaus and Suri, Subhash and Verbeek, Kevin and Willert, Max}, title = {{Tight Bounds for Conflict-Free Chromatic Guarding of Orthogonal Art Galleries}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {421--435}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.421}, URN = {urn:nbn:de:0030-drops-50970}, doi = {10.4230/LIPIcs.SOCG.2015.421}, annote = {Keywords: Orthogonal polygons, art gallery problem, hypergraph coloring} }

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