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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Map matching is a common task when analysing GPS tracks, such as vehicle trajectories. The goal is to match a recorded noisy polygonal curve to a path on the map, usually represented as a geometric graph. The Fréchet distance is a commonly used metric for curves, making it a natural fit. The map-matching problem is well-studied, yet until recently no-one tackled the data structure question: preprocess a given graph so that one can query the minimum Fréchet distance between all graph paths and a polygonal curve. Recently, Gudmundsson, Seybold, and Wong [Gudmundsson et al., 2023] studied this problem for arbitrary query polygonal curves and c-packed graphs. In this paper, we instead require the graphs to be λ-low-density t-spanners, which is significantly more representative of real-world networks. We also show how to report a path that minimises the distance efficiently rather than only returning the minimal distance, which was stated as an open problem in their paper.

Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Aleksandr Popov, and Sampson Wong. Map-Matching Queries Under Fréchet Distance on Low-Density Spanners. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{buchin_et_al:LIPIcs.SoCG.2024.27, author = {Buchin, Kevin and Buchin, Maike and Gudmundsson, Joachim and Popov, Aleksandr and Wong, Sampson}, title = {{Map-Matching Queries Under Fr\'{e}chet Distance on Low-Density Spanners}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {27:1--27:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.27}, URN = {urn:nbn:de:0030-drops-199723}, doi = {10.4230/LIPIcs.SoCG.2024.27}, annote = {Keywords: Map Matching, Fr\'{e}chet Distance, Data Structures} }

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**Published in:** Dagstuhl Reports, Volume 13, Issue 5 (2023)

This report documents the program and the outcomes of the Dagstuhl Seminar 23221 "Computational Geometry". The seminar was held from May 29th to June 2nd, 2023, and 39 participants from various countries attended it, including two remote participants. Recent advances in computational geometry were presented and discussed, and new challenges were identified. This report collects the abstracts of the talks and the open problems presented at the seminar.

Siu-Wing Cheng, Maarten Löffler, Jeff M. Phillips, and Aleksandr Popov. Computational Geometry (Dagstuhl Seminar 23221). In Dagstuhl Reports, Volume 13, Issue 5, pp. 165-181, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{cheng_et_al:DagRep.13.5.165, author = {Cheng, Siu-Wing and L\"{o}ffler, Maarten and Phillips, Jeff M. and Popov, Aleksandr}, title = {{Computational Geometry (Dagstuhl Seminar 23221)}}, pages = {165--181}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2023}, volume = {13}, number = {5}, editor = {Cheng, Siu-Wing and L\"{o}ffler, Maarten and Phillips, Jeff M. and Popov, Aleksandr}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.13.5.165}, URN = {urn:nbn:de:0030-drops-193692}, doi = {10.4230/DagRep.13.5.165}, annote = {Keywords: Algorithms, Combinatorics, Geometric Computing, Reconfiguration, Uncertainty} }

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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 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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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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