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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We consider the problem of testing, for a given set of planar regions R and an integer k, whether there exists a convex shape whose boundary intersects at least k regions of R. We provide polynomial-time algorithms for the case where the regions are disjoint axis-aligned rectangles or disjoint line segments with a constant number of orientations. On the other hand, we show that the problem is NP-hard when the regions are intersecting axis-aligned rectangles or 3-oriented line segments. For several natural intermediate classes of shapes (arbitrary disjoint segments, intersecting 2-oriented segments) the problem remains open.

Vahideh Keikha, Mees van de Kerkhof, Marc van Kreveld, Irina Kostitsyna, Maarten Löffler, Frank Staals, Jérôme Urhausen, Jordi L. Vermeulen, and Lionov Wiratma. Convex Partial Transversals of Planar Regions. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 52:1-52:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{keikha_et_al:LIPIcs.ISAAC.2018.52, author = {Keikha, Vahideh and van de Kerkhof, Mees and van Kreveld, Marc and Kostitsyna, Irina and L\"{o}ffler, Maarten and Staals, Frank and Urhausen, J\'{e}r\^{o}me and Vermeulen, Jordi L. and Wiratma, Lionov}, title = {{Convex Partial Transversals of Planar Regions}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {52:1--52:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.52}, URN = {urn:nbn:de:0030-drops-100003}, doi = {10.4230/LIPIcs.ISAAC.2018.52}, annote = {Keywords: computational geometry, algorithms, NP-hardness, convex transversals} }

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Short Paper

**Published in:** LIPIcs, Volume 114, 10th International Conference on Geographic Information Science (GIScience 2018)

Two of the grouping definitions for trajectories that have been developed in recent years allow a continuous motion model and allow varying shape groups. One of these definitions was suggested as a refinement of the other. In this paper we perform an experimental comparison to highlight the differences in these two definitions on various data sets.

Lionov Wiratma, Maarten Löffler, and Frank Staals. An Experimental Comparison of Two Definitions for Groups of Moving Entities (Short Paper). In 10th International Conference on Geographic Information Science (GIScience 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 114, pp. 64:1-64:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{wiratma_et_al:LIPIcs.GISCIENCE.2018.64, author = {Wiratma, Lionov and L\"{o}ffler, Maarten and Staals, Frank}, title = {{An Experimental Comparison of Two Definitions for Groups of Moving Entities}}, booktitle = {10th International Conference on Geographic Information Science (GIScience 2018)}, pages = {64:1--64:6}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-083-5}, ISSN = {1868-8969}, year = {2018}, volume = {114}, editor = {Winter, Stephan and Griffin, Amy and Sester, Monika}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GISCIENCE.2018.64}, URN = {urn:nbn:de:0030-drops-93928}, doi = {10.4230/LIPIcs.GISCIENCE.2018.64}, annote = {Keywords: Trajectories, grouping algorithms, experimental comparison} }

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

We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given epsilon>0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most epsilon.
We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the algorithms are given a considerably larger error threshold than epsilon. Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard. The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time. Finally, to compute the optimal simplification from a polygonal line consisting of n vertices under the Fréchet distance, we give an O(kn^5) time algorithm that requires O(kn^2) space, where k is the output complexity of the simplification.

Marc van Kreveld, Maarten Löffler, and Lionov Wiratma. On Optimal Polyline Simplification Using the Hausdorff and Fréchet Distance. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{vankreveld_et_al:LIPIcs.SoCG.2018.56, author = {van Kreveld, Marc and L\"{o}ffler, Maarten and Wiratma, Lionov}, title = {{On Optimal Polyline Simplification Using the Hausdorff and Fr\'{e}chet Distance}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {56:1--56:14}, 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.2018.56}, URN = {urn:nbn:de:0030-drops-87694}, doi = {10.4230/LIPIcs.SoCG.2018.56}, annote = {Keywords: polygonal line simplification, Hausdorff distance, Fr\'{e}chet distance, Imai-Iri, Douglas-Peucker} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

One of the important tasks in the analysis of spatio-temporal data collected from moving entities is to find a group: a set of entities that travel together for a sufficiently long period of time. Buchin et al. [JoCG, 2015] introduce a formal definition of groups, analyze its mathematical structure, and present efficient algorithms for computing all maximal groups in a given set of trajectories. In this paper, we refine their definition and argue that our proposed definition corresponds better to human intuition in certain cases, particularly in dense environments.
We present algorithms to compute all maximal groups from a set of moving entities according to the new definition. For a set of n moving entities in R^1, specified by linear interpolation in a sequence of tau time stamps, we show that all maximal groups can be computed in O(tau^2 n^4) time. A similar approach applies if the time stamps of entities are not the same, at the cost of a small extra factor of alpha(n) in the running time. In higher dimensions, we can compute all maximal groups in O(tau^2 n^5 log n) time (for any constant number of dimensions).
We also show that one tau factor can be traded for a much higher dependence on n by giving a O(tau n^4 2^n) algorithm for the same problem. Consequently, we give a linear-time algorithm when the number of entities is constant and the input size relates to the number of time stamps of each entity. Finally, we provide a construction to show that it might be difficult to develop an algorithm with polynomial dependence on n and linear dependence on tau.

Marc van Kreveld, Maarten Löffler, Frank Staals, and Lionov Wiratma. A Refined Definition for Groups of Moving Entities and its Computation. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 48:1-48:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{vankreveld_et_al:LIPIcs.ISAAC.2016.48, author = {van Kreveld, Marc and L\"{o}ffler, Maarten and Staals, Frank and Wiratma, Lionov}, title = {{A Refined Definition for Groups of Moving Entities and its Computation}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {48:1--48:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.48}, URN = {urn:nbn:de:0030-drops-68188}, doi = {10.4230/LIPIcs.ISAAC.2016.48}, annote = {Keywords: moving entities, trajectories, grouping, computational geometry} }