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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Given a point set P in the Euclidean plane and a parameter t, we define an oriented t-spanner as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in G through those points is at most a factor t longer than the shortest oriented cycle in the complete bi-directed graph. We investigate the problem of computing sparse graphs with small oriented dilation.
As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a 1-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in 𝒪(n⁸) time for n points, and a greedy algorithm that computes a 5-spanner in 𝒪(nlog n) time.
Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in an oriented 𝒪(1)-spanner.

Kevin Buchin, Joachim Gudmundsson, Antonia Kalb, Aleksandr Popov, Carolin Rehs, André van Renssen, and Sampson Wong. Oriented Spanners. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 26:1-26:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{buchin_et_al:LIPIcs.ESA.2023.26, author = {Buchin, Kevin and Gudmundsson, Joachim and Kalb, Antonia and Popov, Aleksandr and Rehs, Carolin and van Renssen, Andr\'{e} and Wong, Sampson}, title = {{Oriented Spanners}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {26:1--26:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.26}, URN = {urn:nbn:de:0030-drops-186796}, doi = {10.4230/LIPIcs.ESA.2023.26}, annote = {Keywords: computational geometry, spanner, oriented graph, greedy triangulation} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Let P be a simple polygon with n vertices, and let A be a set of m points or line segments inside P. We develop data structures that can efficiently count the objects from A that are visible to a query point or a query segment. Our main aim is to obtain fast, O(polylog nm), query times, while using as little space as possible.
In case the query is a single point, a simple visibility-polygon-based solution achieves O(log nm) query time using O(nm²) space. In case A also contains only points, we present a smaller, O(n + m^{2+ε} log n)-space, data structure based on a hierarchical decomposition of the polygon.
Building on these results, we tackle the case where the query is a line segment and A contains only points. The main complication here is that the segment may intersect multiple regions of the polygon decomposition, and that a point may see multiple such pieces. Despite these issues, we show how to achieve O(log n log nm) query time using only O(nm^{2+ε} + n²) space. Finally, we show that we can even handle the case where the objects in A are segments with the same bounds.

Kevin Buchin, Bram Custers, Ivor van der Hoog, Maarten Löffler, Aleksandr Popov, Marcel Roeloffzen, and Frank Staals. Segment Visibility Counting Queries in Polygons. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 58:1-58:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{buchin_et_al:LIPIcs.ISAAC.2022.58, author = {Buchin, Kevin and Custers, Bram and van der Hoog, Ivor and L\"{o}ffler, Maarten and Popov, Aleksandr and Roeloffzen, Marcel and Staals, Frank}, title = {{Segment Visibility Counting Queries in Polygons}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {58:1--58:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.58}, URN = {urn:nbn:de:0030-drops-173431}, doi = {10.4230/LIPIcs.ISAAC.2022.58}, annote = {Keywords: Visibility, Data Structure, Polygons, Complexity} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

We consider the following surveillance problem: Given a set P of n sites in a metric space and a set R of k robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency L of a schedule is the maximum latency of any site, where the latency of a site s is the supremum of the lengths of the time intervals between consecutive visits to s.
When k = 1 the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. For k ≥ 2 (which is the version we are interested in) the problem becomes even more challenging; for example, it is not even clear if the decision version of the problem is decidable, in particular in the Euclidean case.
We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into 𝓁 groups, for some 𝓁 ≤ k, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a 1+ε factor loss in the approximation factor and an O((k/ε) ^k) factor loss in the runtime, for any ε > 0. Our second main result shows that an optimal cyclic solution is a 2(1-1/k)-approximation of the overall optimal solution. Note that for k = 2 this implies that an optimal cyclic solution is optimal overall. We conjecture that this is true for k ≥ 3 as well.
The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a (2(1-1/k)+ε)-approximation of the optimal unrestricted (not necessarily cyclic) solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting. Similar results can be obtained by combining our results with other known TSP algorithms in non-Euclidean metrics.

Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang. On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 2:1-2:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{afshani_et_al:LIPIcs.SoCG.2022.2, author = {Afshani, Peyman and de Berg, Mark and Buchin, Kevin and Gao, Jie and L\"{o}ffler, Maarten and Nayyeri, Amir and Raichel, Benjamin and Sarkar, Rik and Wang, Haotian and Yang, Hao-Tsung}, title = {{On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {2:1--2:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.2}, URN = {urn:nbn:de:0030-drops-160109}, doi = {10.4230/LIPIcs.SoCG.2022.2}, annote = {Keywords: Approximation, Motion Planning, Scheduling} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

We consider the unlabeled motion-planning problem of m unit-disc robots moving in a simple polygonal workspace of n edges. The goal is to find a motion plan that moves the robots to a given set of m target positions. For the unlabeled variant, it does not matter which robot reaches which target position as long as all target positions are occupied in the end.
If the workspace has narrow passages such that the robots cannot fit through them, then the free configuration space, representing all possible unobstructed positions of the robots, will consist of multiple connected components. Even if in each component of the free space the number of targets matches the number of start positions, the motion-planning problem does not always have a solution when the robots and their targets are positioned very densely. In this paper, we prove tight bounds on how much separation between start and target positions is necessary to always guarantee a solution. Moreover, we describe an algorithm that always finds a solution in time O(n log n + mn + m²) if the separation bounds are met. Specifically, we prove that the following separation is sufficient: any two start positions are at least distance 4 apart, any two target positions are at least distance 4 apart, and any pair of a start and a target positions is at least distance 3 apart. We further show that when the free space consists of a single connected component, the separation between start and target positions is not necessary.

Bahareh Banyassady, Mark de Berg, Karl Bringmann, Kevin Buchin, Henning Fernau, Dan Halperin, Irina Kostitsyna, Yoshio Okamoto, and Stijn Slot. Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 12:1-12:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{banyassady_et_al:LIPIcs.SoCG.2022.12, author = {Banyassady, Bahareh and de Berg, Mark and Bringmann, Karl and Buchin, Kevin and Fernau, Henning and Halperin, Dan and Kostitsyna, Irina and Okamoto, Yoshio and Slot, Stijn}, title = {{Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {12:1--12:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.12}, URN = {urn:nbn:de:0030-drops-160203}, doi = {10.4230/LIPIcs.SoCG.2022.12}, annote = {Keywords: motion planning, computational geometry, simple polygon} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Dynamic Time Warping is arguably the most popular similarity measure for time series, where we define a time series to be a one-dimensional polygonal curve. The drawback of Dynamic Time Warping is that it is sensitive to the sampling rate of the time series. The Fréchet distance is an alternative that has gained popularity, however, its drawback is that it is sensitive to outliers.
Continuous Dynamic Time Warping (CDTW) is a recently proposed alternative that does not exhibit the aforementioned drawbacks. CDTW combines the continuous nature of the Fréchet distance with the summation of Dynamic Time Warping, resulting in a similarity measure that is robust to sampling rate and to outliers. In a recent experimental work of Brankovic et al., it was demonstrated that clustering under CDTW avoids the unwanted artifacts that appear when clustering under Dynamic Time Warping and under the Fréchet distance. Despite its advantages, the major shortcoming of CDTW is that there is no exact algorithm for computing CDTW, in polynomial time or otherwise.
In this work, we present the first exact algorithm for computing CDTW of one-dimensional curves. Our algorithm runs in time 𝒪(n⁵) for a pair of one-dimensional curves, each with complexity at most n. In our algorithm, we propagate continuous functions in the dynamic program for CDTW, where the main difficulty lies in bounding the complexity of the functions. We believe that our result is an important first step towards CDTW becoming a practical similarity measure between curves.

Kevin Buchin, André Nusser, and Sampson Wong. Computing Continuous Dynamic Time Warping of Time Series in Polynomial Time. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 22:1-22:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2022.22, author = {Buchin, Kevin and Nusser, Andr\'{e} and Wong, Sampson}, title = {{Computing Continuous Dynamic Time Warping of Time Series in Polynomial Time}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {22:1--22:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.22}, URN = {urn:nbn:de:0030-drops-160307}, doi = {10.4230/LIPIcs.SoCG.2022.22}, annote = {Keywords: Computational Geometry, Curve Similarity, Fr\'{e}chet distance, Dynamic Time Warping, Continuous Dynamic Time Warping} }

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

Exactly 20 years ago at MFCS, Demaine posed the open problem whether the game of Dots & Boxes is PSPACE-complete. Dots & Boxes has been studied extensively, with for instance a chapter in Berlekamp et al. Winning Ways for Your Mathematical Plays, a whole book on the game The Dots and Boxes Game: Sophisticated Child’s Play by Berlekamp, and numerous articles in the Games of No Chance series. While known to be NP-hard, the question of its complexity remained open. We resolve this question, proving that the game is PSPACE-complete by a reduction from a game played on propositional formulas.

Kevin Buchin, Mart Hagedoorn, Irina Kostitsyna, and Max van Mulken. Dots & Boxes Is PSPACE-Complete. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 25:1-25:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{buchin_et_al:LIPIcs.MFCS.2021.25, author = {Buchin, Kevin and Hagedoorn, Mart and Kostitsyna, Irina and van Mulken, Max}, title = {{Dots \& Boxes Is PSPACE-Complete}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {25:1--25:18}, 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.2021.25}, URN = {urn:nbn:de:0030-drops-144657}, doi = {10.4230/LIPIcs.MFCS.2021.25}, annote = {Keywords: Dots \& Boxes, PSPACE-complete, combinatorial game} }

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

We study the problem of polygonal curve simplification under uncertainty, where instead of a sequence of exact points, each uncertain point is represented by a region which contains the (unknown) true location of the vertex. The regions we consider are disks, line segments, convex polygons, and discrete sets of points. We are interested in finding the shortest subsequence of uncertain points such that no matter what the true location of each uncertain point is, the resulting polygonal curve is a valid simplification of the original polygonal curve under the Hausdorff or the Fréchet distance. For both these distance measures, we present polynomial-time algorithms for this problem.

Kevin Buchin, Maarten Löffler, Aleksandr Popov, and Marcel Roeloffzen. Uncertain Curve Simplification. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 26:1-26:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{buchin_et_al:LIPIcs.MFCS.2021.26, author = {Buchin, Kevin and L\"{o}ffler, Maarten and Popov, Aleksandr and Roeloffzen, Marcel}, title = {{Uncertain Curve Simplification}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {26:1--26:22}, 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.2021.26}, URN = {urn:nbn:de:0030-drops-144666}, doi = {10.4230/LIPIcs.MFCS.2021.26}, annote = {Keywords: Curves, Uncertainty, Simplification, Fr\'{e}chet Distance, Hausdorff Distance} }

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Complete Volume

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

LIPIcs, Volume 189, SoCG 2021, Complete Volume

Kevin Buchin and Éric Colin de Verdière. LIPIcs, Volume 189, SoCG 2021, Complete Volume. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 1-978, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@Proceedings{buchin_et_al:LIPIcs.SoCG.2021, title = {{LIPIcs, Volume 189, SoCG 2021, Complete Volume}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {1--978}, 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}, URN = {urn:nbn:de:0030-drops-137987}, doi = {10.4230/LIPIcs.SoCG.2021}, annote = {Keywords: LIPIcs, Volume 189, SoCG 2021, Complete Volume} }

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Front Matter

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

Front Matter, Table of Contents, Preface, Conference Organization

Kevin Buchin and Éric Colin de Verdière. Front Matter, Table of Contents, Preface, Conference Organization. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 0:i-0:xviii, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2021.0, author = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {0:i--0:xviii}, 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.2021.0}, URN = {urn:nbn:de:0030-drops-137993}, doi = {10.4230/LIPIcs.SoCG.2021.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs, we are given a graph G that is embedded in Euclidean space. The edges of G need to be scanned, i.e., probed from both of their vertices. In order to scan their edge, two vertices need to face each other; changing the heading of a vertex incurs some cost in terms of energy or rotation time that is proportional to the corresponding rotation angle. Our goal is to compute schedules that minimize the following objective functions: (i) in Minimum Makespan Scan Cover (MSC-MS), this is the time until all edges are scanned; (ii) in Minimum Total Energy Scan Cover (MSC-TE), the sum of all rotation angles; (iii) in Minimum Bottleneck Energy Scan Cover (MSC-BE), the maximum total rotation angle at one vertex.
Previous theoretical work on MSC-MS revealed a close connection to graph coloring and the cut cover problem, leading to hardness and approximability results. In this paper, we present polynomial-time algorithms for 1D instances of MSC-TE and MSC-BE, but NP-hardness proofs for bipartite 2D instances. For bipartite graphs in 2D, we also give 2-approximation algorithms for both MSC-TE and MSC-BE. Most importantly, we provide a comprehensive study of practical methods for all three problems. We compare three different mixed-integer programming and two constraint programming approaches, and show how to compute provably optimal solutions for geometric instances with up to 300 edges. Additionally, we compare the performance of different meta-heuristics for even larger instances.

Kevin Buchin, Sándor P. Fekete, Alexander Hill, Linda Kleist, Irina Kostitsyna, Dominik Krupke, Roel Lambers, and Martijn Struijs. Minimum Scan Cover and Variants - Theory and Experiments. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 4:1-4:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{buchin_et_al:LIPIcs.SEA.2021.4, author = {Buchin, Kevin and Fekete, S\'{a}ndor P. and Hill, Alexander and Kleist, Linda and Kostitsyna, Irina and Krupke, Dominik and Lambers, Roel and Struijs, Martijn}, title = {{Minimum Scan Cover and Variants - Theory and Experiments}}, booktitle = {19th International Symposium on Experimental Algorithms (SEA 2021)}, pages = {4:1--4:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-185-6}, ISSN = {1868-8969}, year = {2021}, volume = {190}, editor = {Coudert, David and Natale, Emanuele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.4}, URN = {urn:nbn:de:0030-drops-137765}, doi = {10.4230/LIPIcs.SEA.2021.4}, annote = {Keywords: Graph scanning, angular metric, makespan, energy, bottleneck, complexity, approximation, algorithm engineering, mixed-integer programming, constraint programming} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Reliable spanners can withstand huge failures, even when a linear number of vertices are deleted from the network. In case of failures, some of the remaining vertices of a reliable spanner may no longer admit the spanner property, but this collateral damage is bounded by a fraction of the size of the attack. It is known that Ω(nlog n) edges are needed to achieve this strong property, where n is the number of vertices in the network, even in one dimension. Constructions of reliable geometric (1+ε)-spanners, for n points in ℝ^d, are known, where the resulting graph has 𝒪(n log n log log⁶n) edges.
Here, we show randomized constructions of smaller size spanners that have the desired reliability property in expectation or with good probability. The new construction is simple, and potentially practical - replacing a hierarchical usage of expanders (which renders the previous constructions impractical) by a simple skip list like construction. This results in a 1-spanner, on the line, that has linear number of edges. Using this, we present a construction of a reliable spanner in ℝ^d with 𝒪(n log log²n log log log n) edges.

Kevin Buchin, Sariel Har-Peled, and Dániel Oláh. Sometimes Reliable Spanners of Almost Linear Size. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 27:1-27:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{buchin_et_al:LIPIcs.ESA.2020.27, author = {Buchin, Kevin and Har-Peled, Sariel and Ol\'{a}h, D\'{a}niel}, title = {{Sometimes Reliable Spanners of Almost Linear Size}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {27:1--27:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.27}, URN = {urn:nbn:de:0030-drops-128934}, doi = {10.4230/LIPIcs.ESA.2020.27}, annote = {Keywords: Geometric spanners, vertex failures, reliability} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In this paper we study a wide range of variants for computing the (discrete and continuous) Fréchet distance between uncertain curves. We define an uncertain curve as a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound Fréchet distance, which are the minimum and maximum Fréchet distance for any realisations of the curves.
We prove that both problems are NP-hard for the continuous Fréchet distance, and the upper bound problem remains hard for the discrete Fréchet distance. In contrast, the lower bound discrete Fréchet distance can be computed in polynomial time using dynamic programming. Furthermore, we show that computing the expected discrete or continuous Fréchet distance is #P-hard when the uncertainty regions are modelled as point sets or line segments.
On the positive side, we argue that in any constant dimension there is a FPTAS for the lower bound problem when Δ/δ is polynomially bounded, where δ is the Fréchet distance and Δ bounds the diameter of the regions. We then argue there is a near-linear-time 3-approximation for the decision problem when the regions are convex and roughly δ-separated. Finally, we study the setting with Sakoe - Chiba bands, restricting the alignment of the two curves, and give polynomial-time algorithms for upper bound and expected (discrete) Fréchet distance for point-set-modelled uncertainty regions.

Kevin Buchin, Chenglin Fan, Maarten Löffler, Aleksandr Popov, Benjamin Raichel, and Marcel Roeloffzen. Fréchet Distance for Uncertain Curves. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 20:1-20:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{buchin_et_al:LIPIcs.ICALP.2020.20, author = {Buchin, Kevin and Fan, Chenglin and L\"{o}ffler, Maarten and Popov, Aleksandr and Raichel, Benjamin and Roeloffzen, Marcel}, title = {{Fr\'{e}chet Distance for Uncertain Curves}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {20:1--20:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.20}, URN = {urn:nbn:de:0030-drops-124276}, doi = {10.4230/LIPIcs.ICALP.2020.20}, annote = {Keywords: Curves, Uncertainty, Fr\'{e}chet Distance, Hardness} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

We study the complexity of clustering curves under k-median and k-center objectives in the metric space of the Fréchet distance and related distance measures. Building upon recent hardness results for the minimum-enclosing-ball problem under the Fréchet distance, we show that also the 1-median problem is NP-hard. Furthermore, we show that the 1-median problem is W[1]-hard with the number of curves as parameter. We show this under the discrete and continuous Fréchet and Dynamic Time Warping (DTW) distance. This yields an independent proof of an earlier result by Bulteau et al. from 2018 for a variant of DTW that uses squared distances, where the new proof is both simpler and more general. On the positive side, we give approximation algorithms for problem variants where the center curve may have complexity at most 𝓁 under the discrete Fréchet distance. In particular, for fixed k, 𝓁 and ε, we give (1+ε)-approximation algorithms for the (k,𝓁)-median and (k,𝓁)-center objectives and a polynomial-time exact algorithm for the (k,𝓁)-center objective.

Kevin Buchin, Anne Driemel, and Martijn Struijs. On the Hardness of Computing an Average Curve. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 19:1-19:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{buchin_et_al:LIPIcs.SWAT.2020.19, author = {Buchin, Kevin and Driemel, Anne and Struijs, Martijn}, title = {{On the Hardness of Computing an Average Curve}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {19:1--19:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.19}, URN = {urn:nbn:de:0030-drops-122662}, doi = {10.4230/LIPIcs.SWAT.2020.19}, annote = {Keywords: Curves, Clustering, Algorithms, Hardness, Approximation} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

We consider paths with low exposure to a 2D polygonal domain, i.e., paths which are seen as little as possible; we differentiate between integral exposure (when we care about how long the path sees every point of the domain) and 0/1 exposure (just counting whether a point is seen by the path or not). For the integral exposure, we give a PTAS for finding the minimum-exposure path between two given points in the domain; for the 0/1 version, we prove that in a simple polygon the shortest path has the minimum exposure, while in domains with holes the problem becomes NP-hard. We also highlight connections of the problem to minimum satisfiability and settle hardness of variants of planar min- and max-SAT.

Kevin Buchin, Valentin Polishchuk, Leonid Sedov, and Roman Voronov. Geometric Secluded Paths and Planar Satisfiability. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 24:1-24:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2020.24, author = {Buchin, Kevin and Polishchuk, Valentin and Sedov, Leonid and Voronov, Roman}, title = {{Geometric Secluded Paths and Planar Satisfiability}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.24}, URN = {urn:nbn:de:0030-drops-121827}, doi = {10.4230/LIPIcs.SoCG.2020.24}, annote = {Keywords: Visibility, Route planning, Security/privacy, Planar satisfiability} }

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Media Exposition

**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

We present a new game, Dots & Polygons, played on a planar point set. We prove that its NP-hard and discuss strategies for the case when the point set is in convex position.

Kevin Buchin, Mart Hagedoorn, Irina Kostitsyna, Max van Mulken, Jolan Rensen, and Leo van Schooten. Dots & Polygons (Media Exposition). In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 79:1-79:4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2020.79, author = {Buchin, Kevin and Hagedoorn, Mart and Kostitsyna, Irina and van Mulken, Max and Rensen, Jolan and van Schooten, Leo}, title = {{Dots \& Polygons}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {79:1--79:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.79}, URN = {urn:nbn:de:0030-drops-122371}, doi = {10.4230/LIPIcs.SoCG.2020.79}, annote = {Keywords: Dots \& Boxes, NP-hard, game, cycle packing} }

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Media Exposition

**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

We present a method for generating interesting levels based on several NP-hardness reductions for a puzzle game based on the Art Gallery problem.

Toon van Benthem, Kevin Buchin, Irina Kostitsyna, and Stijn Slot. Designing Art Galleries (Media Exposition). In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 80:1-80:5, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{vanbenthem_et_al:LIPIcs.SoCG.2020.80, author = {van Benthem, Toon and Buchin, Kevin and Kostitsyna, Irina and Slot, Stijn}, title = {{Designing Art Galleries}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {80:1--80:5}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.80}, URN = {urn:nbn:de:0030-drops-122382}, doi = {10.4230/LIPIcs.SoCG.2020.80}, annote = {Keywords: Art Gallery problem, NP-hard, puzzle, level generation} }

Document

**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

We show how to construct (1+epsilon)-spanner over a set P of n points in R^d that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters theta, epsilon in (0,1), the computed spanner G has O(epsilon^{-7d} log^7 epsilon^{-1} * theta^{-6} n log n (log log n)^6) edges. Furthermore, for any k, and any deleted set B subseteq P of k points, the residual graph G \ B is (1+epsilon)-spanner for all the points of P except for (1+theta)k of them. No previous constructions, beyond the trivial clique with O(n^2) edges, were known such that only a tiny additional fraction (i.e., theta) lose their distance preserving connectivity.
Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one dimensional construction in a black box fashion.

Kevin Buchin, Sariel Har-Peled, and Dániel Oláh. A Spanner for the Day After. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 19:1-19:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2019.19, author = {Buchin, Kevin and Har-Peled, Sariel and Ol\'{a}h, D\'{a}niel}, title = {{A Spanner for the Day After}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {19:1--19:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.19}, URN = {urn:nbn:de:0030-drops-104237}, doi = {10.4230/LIPIcs.SoCG.2019.19}, annote = {Keywords: Geometric spanners, vertex failures, robustness} }

Document

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

Robust estimators, like the median of a point set, are important for data analysis in the presence of outliers. We study robust estimators for locationally uncertain points with discrete distributions. That is, each point in a data set has a discrete probability distribution describing its location. The probabilistic nature of uncertain data makes it challenging to compute such estimators, since the true value of the estimator is now described by a distribution rather than a single point. We show how to construct and estimate the distribution of the median of a point set. Building the approximate support of the distribution takes near-linear time, and assigning probability to that support takes quadratic time. We also develop a general approximation technique for distributions of robust estimators with respect to ranges with bounded VC dimension. This includes the geometric median for high dimensions and the Siegel estimator for linear regression.

Kevin Buchin, Jeff M. Phillips, and Pingfan Tang. Approximating the Distribution of the Median and other Robust Estimators on Uncertain Data. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 16:1-16:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2018.16, author = {Buchin, Kevin and Phillips, Jeff M. and Tang, Pingfan}, title = {{Approximating the Distribution of the Median and other Robust Estimators on Uncertain Data}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {16:1--16:14}, 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.2018.16}, URN = {urn:nbn:de:0030-drops-87292}, doi = {10.4230/LIPIcs.SoCG.2018.16}, annote = {Keywords: Uncertain Data, Robust Estimators, Geometric Median, Tukey Median} }

Document

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

We consider the problem of packing a family of disks 'on a shelf,'
that is, such that each disk touches the x-axis from above and such that no two disks overlap. We prove that the problem of minimizing the distance between the leftmost point and the rightmost point of any disk is NP-hard. On the positive side, we show how to approximate this problem within a factor of 4/3 in O(n log n) time, and provide an O(n log n)-time exact algorithm for a special case, in particular when the ratio between the largest and smallest radius is at most four.

Helmut Alt, Kevin Buchin, Steven Chaplick, Otfried Cheong, Philipp Kindermann, Christian Knauer, and Fabian Stehn. Placing your Coins on a Shelf. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 4:1-4:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{alt_et_al:LIPIcs.ISAAC.2017.4, author = {Alt, Helmut and Buchin, Kevin and Chaplick, Steven and Cheong, Otfried and Kindermann, Philipp and Knauer, Christian and Stehn, Fabian}, title = {{Placing your Coins on a Shelf}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {4:1--4:12}, 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.2017.4}, URN = {urn:nbn:de:0030-drops-82145}, doi = {10.4230/LIPIcs.ISAAC.2017.4}, annote = {Keywords: packing problems, approximation algorithms, NP-hardness} }

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

Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P_1 and P_2 such that the sum of the perimeters of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes the convex hull of P_i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n^2) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(n log^4 n) time and a (1+e)-approximation algorithm running in O(n + 1/e^2 log^4(1/e)) time.

Mikkel Abrahamsen, Mark de Berg, Kevin Buchin, Mehran Mehr, and Ali D. Mehrabi. Minimum Perimeter-Sum Partitions in the Plane. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 4:1-4:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2017.4, author = {Abrahamsen, Mikkel and de Berg, Mark and Buchin, Kevin and Mehr, Mehran and Mehrabi, Ali D.}, title = {{Minimum Perimeter-Sum Partitions in the Plane}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {4:1--4:15}, 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.2017.4}, URN = {urn:nbn:de:0030-drops-72048}, doi = {10.4230/LIPIcs.SoCG.2017.4}, annote = {Keywords: Computational geometry, clustering, minimum-perimeter partition, convex hull} }

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

In a geometric k-clustering problem the goal is to partition a set of points in R^d into k subsets such that a certain cost function of the clustering is minimized. We present data structures for orthogonal range-clustering queries on a point set S: given a query box Q and an integer k > 2, compute an optimal k-clustering for the subset of S inside Q. We obtain the following results.
* We present a general method to compute a (1+epsilon)-approximation to a range-clustering query, where epsilon>0 is a parameter that can be specified as part of the query. Our method applies to a large class of clustering problems, including k-center clustering in any Lp-metric and a variant of k-center clustering where the goal is to minimize the sum (instead of maximum) of the cluster sizes.
* We extend our method to deal with capacitated k-clustering problems, where each of the clusters should not contain more than a given number of points.
* For the special cases of rectilinear k-center clustering in R^1, and in R^2 for k = 2 or 3, we present data structures that answer range-clustering queries exactly.

Mikkel Abrahamsen, Mark de Berg, Kevin Buchin, Mehran Mehr, and Ali D. Mehrabi. Range-Clustering Queries. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 5:1-5:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2017.5, author = {Abrahamsen, Mikkel and de Berg, Mark and Buchin, Kevin and Mehr, Mehran and Mehrabi, Ali D.}, title = {{Range-Clustering Queries}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {5:1--5:16}, 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.2017.5}, URN = {urn:nbn:de:0030-drops-72147}, doi = {10.4230/LIPIcs.SoCG.2017.5}, annote = {Keywords: Geometric data structures, clustering, k-center problem} }

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Multimedia Contribution

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

Ruler of the Plane is a set of games illustrating concepts from combinatorial and computational geometry. The games are based on the art gallery problem, ham-sandwich cuts, the Voronoi game, and geometric network connectivity problems like the Euclidean minimum spanning tree and traveling salesperson problem.

Sander Beekhuis, Kevin Buchin, Thom Castermans, Thom Hurks, and Willem Sonke. Ruler of the Plane - Games of Geometry (Multimedia Contribution). In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 63:1-63:5, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{beekhuis_et_al:LIPIcs.SoCG.2017.63, author = {Beekhuis, Sander and Buchin, Kevin and Castermans, Thom and Hurks, Thom and Sonke, Willem}, title = {{Ruler of the Plane - Games of Geometry}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {63:1--63:5}, 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.2017.63}, URN = {urn:nbn:de:0030-drops-72400}, doi = {10.4230/LIPIcs.SoCG.2017.63}, annote = {Keywords: art gallery problem, ham-sandwich cuts, Voronoi game, traveling sales-person problem} }

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Multimedia Contribution

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

By folding the free-space diagram for efficient preprocessing, we show that the Frechet distance between 1D curves can be computed in O(nk log n) time, assuming one curve has ply k.

Kevin Buchin, Jinhee Chun, Maarten Löffler, Aleksandar Markovic, Wouter Meulemans, Yoshio Okamoto, and Taichi Shiitada. Folding Free-Space Diagrams: Computing the Fréchet Distance between 1-Dimensional Curves (Multimedia Contribution). In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 64:1-64:5, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2017.64, author = {Buchin, Kevin and Chun, Jinhee and L\"{o}ffler, Maarten and Markovic, Aleksandar and Meulemans, Wouter and Okamoto, Yoshio and Shiitada, Taichi}, title = {{Folding Free-Space Diagrams: Computing the Fr\'{e}chet Distance between 1-Dimensional Curves}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {64:1--64:5}, 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.2017.64}, URN = {urn:nbn:de:0030-drops-72417}, doi = {10.4230/LIPIcs.SoCG.2017.64}, annote = {Keywords: Frechet distance, ply, k-packed curves} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity.
Our first set of results is motivated by the Bitonic tsp problem: given a set of n points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamicprogramming exercise to solve this problem in O(n^2) time. While the near-quadratic dependency of similar dynamic programs for Longest Common Subsequence and Discrete Fréchet Distance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic tsp in O(n*log^2(n)) time and its bottleneck version in O(n*log^3(n)) time. In the more general pyramidal tsp problem, the points to be visited are labeled 1, ..., n and the sequence of labels in the solution is required to have at most one local maximum. Our algorithms for the bitonic (bottleneck) tsp problem also work for the pyramidal tsp problem in the plane.
Our second set of results concerns the popular k-opt heuristic for tsp in the graph setting. More precisely, we study the k-opt decision problem, which asks whether a given tour can be improved by a k-opt move that replaces k edges in the tour by k new edges. A simple algorithm solves k-opt in O(n^k) time for fixed k. For 2-opt, this is easily seen to be optimal. For k = 3 we prove that an algorithm with a runtime of the form ~O(n^{3-epsilon}) exists if and only if All-Pairs Shortest Paths in weighted digraphs has such an algorithm. For general k-opt, it is known that a runtime of f(k)*n^{o(k/log(k))} would contradict the Exponential Time Hypothesis. The results for k = 2, 3 may suggest that the actual time complexity of k-opt is Theta(n^k). We show that this is not the case, by presenting an algorithm that finds the best k-move in O(n^{lfoor 2k/3 rfloor +1}) time for fixed k >= 3. This implies that 4-opt can be solved in O(n^3) time, matching the best-known algorithm for 3-opt. Finally, we show how to beat the quadratic barrier for k = 2 in two important settings, namely for points in the plane and when we want to solve 2-opt repeatedly

Mark de Berg, Kevin Buchin, Bart M. P. Jansen, and Gerhard Woeginger. Fine-Grained Complexity Analysis of Two Classic TSP Variants. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 5:1-5:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{deberg_et_al:LIPIcs.ICALP.2016.5, author = {de Berg, Mark and Buchin, Kevin and Jansen, Bart M. P. and Woeginger, Gerhard}, title = {{Fine-Grained Complexity Analysis of Two Classic TSP Variants}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.5}, URN = {urn:nbn:de:0030-drops-62770}, doi = {10.4230/LIPIcs.ICALP.2016.5}, annote = {Keywords: Traveling salesman problem, fine-grained complexity, bitonic tours, k-opt} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10491, Representation, Analysis and Visualization of Moving Objects (2011)

A classification of gull behaviour was produced by the group, led by domain
expert Emiel van Loon, who provided additional context including that gull trips
are typically composed of distinct segments, that gull trips are rarely single
purpose, and that there is very little diurnal pattern to activities. The
classification produced is not intended to be complete, or non overlapping.
Furthermore, the group considered how the attributes in the gulls dataset could be used in algorithms to automatically classify the dataset into distinct spatial
patterns, and associate this with gull behaviours.

Emiel van Loon, Jörg-Rüdiger Sack, Kevin Buchin, Maike Buchin, Mark de Berg, Marc van Kreveld, Joachim Gudmundsson, and David Mountain. 10491 Results of the break-out group: Gulls Data. In Representation, Analysis and Visualization of Moving Objects. Dagstuhl Seminar Proceedings, Volume 10491, pp. 1-4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{vanloon_et_al:DagSemProc.10491.5, author = {van Loon, Emiel and Sack, J\"{o}rg-R\"{u}diger and Buchin, Kevin and Buchin, Maike and de Berg, Mark and van Kreveld, Marc and Gudmundsson, Joachim and Mountain, David}, title = {{10491 Results of the break-out group: Gulls Data}}, booktitle = {Representation, Analysis and Visualization of Moving Objects}, pages = {1--4}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2011}, volume = {10491}, editor = {J\"{o}rg-R\"{u}diger Sack and Bettina Speckmann and Emiel Van Loon and Robert Weibel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10491.5}, URN = {urn:nbn:de:0030-drops-29912}, doi = {10.4230/DagSemProc.10491.5}, annote = {Keywords: Movement classification, Trajectory segmentation} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10491, Representation, Analysis and Visualization of Moving Objects (2011)

Discussions in this group focused on a particular problem that arises in animal
movement ecology: how to link data describing movement (i.e. sequential GPS-
coordinates collected on wild and free-ranging animals) with geographical and
environmental context (i.e. properties of the internal and external environment within which the animals move). Our case study comprised a spatio-temporal data set on the movement of a group of vervet monkeys (Cercopithecus aethiops) over a twelve months observation period. We focused on two topics: context-aware estimation of home range area and multivariate visualisation of context data.

Erik P. Willems, Kevin Buchin, and Urska Demsar. 10491 Results of the break-out group: Movement Data of Vervet Monkeys. In Representation, Analysis and Visualization of Moving Objects. Dagstuhl Seminar Proceedings, Volume 10491, pp. 1-3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{willems_et_al:DagSemProc.10491.6, author = {Willems, Erik P. and Buchin, Kevin and Demsar, Urska}, title = {{10491 Results of the break-out group: Movement Data of Vervet Monkeys}}, booktitle = {Representation, Analysis and Visualization of Moving Objects}, pages = {1--3}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2011}, volume = {10491}, editor = {J\"{o}rg-R\"{u}diger Sack and Bettina Speckmann and Emiel Van Loon and Robert Weibel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10491.6}, URN = {urn:nbn:de:0030-drops-29908}, doi = {10.4230/DagSemProc.10491.6}, annote = {Keywords: Visualisation} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10491, Representation, Analysis and Visualization of Moving Objects (2011)

In this group we decided to collect literature that we were familiar with that best illustrated how movement data in the form of trajectories can be visualised.
In this report we categorise methods based on what part of the data space is shown, i.e. geographical space, temporal space or attribute space, some combination thereof or an aggregation in one or more of the space components. Methods that use computational methods for pattern recognition in combination with visual methods form a separate category. However, these categories are only what we came up with during our short discussion and are therefore not fixed, nor are they mutually exclusive (i.e. there is certain overlap of methods) and should be extended/redefined as required in a more exhaustive literature review in the future.

Kevin Buchin, Urska Demsar, Aidan Slingsby, and Erik P. Willems. 10491 Results of the break-out group: Visualisation. In Representation, Analysis and Visualization of Moving Objects. Dagstuhl Seminar Proceedings, Volume 10491, pp. 1-4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{buchin_et_al:DagSemProc.10491.8, author = {Buchin, Kevin and Demsar, Urska and Slingsby, Aidan and Willems, Erik P.}, title = {{10491 Results of the break-out group: Visualisation}}, booktitle = {Representation, Analysis and Visualization of Moving Objects}, pages = {1--4}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2011}, volume = {10491}, editor = {J\"{o}rg-R\"{u}diger Sack and Bettina Speckmann and Emiel Van Loon and Robert Weibel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10491.8}, URN = {urn:nbn:de:0030-drops-29864}, doi = {10.4230/DagSemProc.10491.8}, annote = {Keywords: Visualisation} }

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