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Documents authored by Buchin, Maike

Document
DOI: 10.4230/LIPIcs.ESA.2024.36
Bicriteria Approximation for Minimum Dilation Graph Augmentation

Authors: Kevin Buchin, Maike Buchin, Joachim Gudmundsson, and Sampson Wong

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
Spanner constructions focus on the initial design of the network. However, networks tend to improve over time. In this paper, we focus on the improvement step. Given a graph and a budget k, which k edges do we add to the graph to minimise its dilation? Gudmundsson and Wong [TALG'22] provided the first positive result for this problem, but their approximation factor is linear in k. Our main result is a (2 √[r]{2} k^{1/r},2r)-bicriteria approximation that runs in O(n³ log n) time, for all r ≥ 1. In other words, if t^* is the minimum dilation after adding any k edges to a graph, then our algorithm adds O(k^{1+1/r}) edges to the graph to obtain a dilation of 2rt^*. Moreover, our analysis of the algorithm is tight under the Erdős girth conjecture.
Cite as

Kevin Buchin, Maike Buchin, Joachim Gudmundsson, and Sampson Wong. Bicriteria Approximation for Minimum Dilation Graph Augmentation. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{buchin_et_al:LIPIcs.ESA.2024.36,
  author =	{Buchin, Kevin and Buchin, Maike and Gudmundsson, Joachim and Wong, Sampson},
  title =	{{Bicriteria Approximation for Minimum Dilation Graph Augmentation}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{36:1--36:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.36},
  URN =		{urn:nbn:de:0030-drops-211079},
  doi =		{10.4230/LIPIcs.ESA.2024.36},
  annote =	{Keywords: Greedy spanner, Graph augmentation}
}
Document
DOI: 10.4230/DagRep.14.2.120
Triangulations in Geometry and Topology (Dagstuhl Seminar 24072)

Authors: Maike Buchin, Jean Cardinal, Arnaud de Mesmay, Jonathan Spreer, and Alex He

Published in: Dagstuhl Reports, Volume 14, Issue 2 (2024)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar "Triangulations in Geometry and Topology" (24072). The seminar was held from February 12 to February 16, 2024, gathered 31 participants, and started with four introductory talks and an open problem session. Then the participants spread into small groups to work on open problems on diverse topics including reconfiguration of geometric shapes, geodesics on triangulated surfaces, distances in flip graphs, geometric cycles and algorithms in 3-manifold topology.
Cite as

Maike Buchin, Jean Cardinal, Arnaud de Mesmay, Jonathan Spreer, and Alex He. Triangulations in Geometry and Topology (Dagstuhl Seminar 24072). In Dagstuhl Reports, Volume 14, Issue 2, pp. 120-163, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{buchin_et_al:DagRep.14.2.120,
  author =	{Buchin, Maike and Cardinal, Jean and de Mesmay, Arnaud and Spreer, Jonathan and He, Alex},
  title =	{{Triangulations in Geometry and Topology (Dagstuhl Seminar 24072)}},
  pages =	{120--163},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2024},
  volume =	{14},
  number =	{2},
  editor =	{Buchin, Maike and Cardinal, Jean and de Mesmay, Arnaud and Spreer, Jonathan and He, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.14.2.120},
  URN =		{urn:nbn:de:0030-drops-205017},
  doi =		{10.4230/DagRep.14.2.120},
  annote =	{Keywords: computational geometry, geometric topology, triangulations}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.27
Map-Matching Queries Under Fréchet Distance on Low-Density Spanners

Authors: Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Aleksandr Popov, and Sampson Wong

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

Abstract
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.
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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}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2024.18
Dynamic L-Budget Clustering of Curves

Authors: Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Lukas Plätz, Lea Thiel, and Sampson Wong

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Abstract
A key goal of clustering is data reduction. In center-based clustering of complex objects therefore not only the number of clusters but also the complexity of the centers plays a crucial role. We propose L-Budget Clustering as unifying perspective on this task, optimizing the clustering under the constraint that the summed complexity of all centers is at most L. We present algorithms for clustering planar curves under the Fréchet distance, but note that our algorithms more generally apply to objects in metric spaces if a notion of simplification of objects is applicable. A scenario in which data reduction is of particular importance is when the space is limited. Our main result is an efficient (8 + ε)-approximation algorithm with a (1 + ε)-resource augmentation that maintains an L-budget clustering under insertion of curves using only O(Lε^{-1}) space and O^*(L³log(L) + L²log(r^*/r₀)) time where O^* hides factors of ε^{-1}.
Cite as

Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Lukas Plätz, Lea Thiel, and Sampson Wong. Dynamic L-Budget Clustering of Curves. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{buchin_et_al:LIPIcs.SWAT.2024.18,
  author =	{Buchin, Kevin and Buchin, Maike and Gudmundsson, Joachim and Pl\"{a}tz, Lukas and Thiel, Lea and Wong, Sampson},
  title =	{{Dynamic L-Budget Clustering of Curves}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.18},
  URN =		{urn:nbn:de:0030-drops-200588},
  doi =		{10.4230/LIPIcs.SWAT.2024.18},
  annote =	{Keywords: clustering, streaming algorithm, polygonal curves, Fr\'{e}chet distance, storage efficiency, simplification, approximation algorithms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.3
Realizability of Free Spaces of Curves

Authors: Hugo A. Akitaya, Maike Buchin, Majid Mirzanezhad, Leonie Ryvkin, and Carola Wenk

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
The free space diagram is a popular tool to compute the well-known Fréchet distance. As the Fréchet distance is used in many different fields, many variants have been established to cover the specific needs of these applications. Often the question arises whether a certain pattern in the free space diagram is realizable, i.e., whether there exists a pair of polygonal chains whose free space diagram corresponds to it. The answer to this question may help in deciding the computational complexity of these distance measures, as well as allowing to design more efficient algorithms for restricted input classes that avoid certain free space patterns. Therefore we study the inverse problem: Given a potential free space diagram, do there exist curves that generate this diagram? Our problem of interest is closely tied to the classic Distance Geometry problem. We settle the complexity of Distance Geometry in ℝ^{>2}, showing ∃ℝ-hardness. We use this to show that for curves in ℝ^{≥2} the realizability problem is ∃ℝ-complete, both for continuous and for discrete Fréchet distance. We prove that the continuous case in ℝ¹ is only weakly NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is fixed-parameter tractable. Interestingly, for the discrete case in ℝ¹ we show that the problem becomes solvable in polynomial time.
Cite as

Hugo A. Akitaya, Maike Buchin, Majid Mirzanezhad, Leonie Ryvkin, and Carola Wenk. Realizability of Free Spaces of Curves. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{a.akitaya_et_al:LIPIcs.ISAAC.2023.3,
  author =	{A. Akitaya, Hugo and Buchin, Maike and Mirzanezhad, Majid and Ryvkin, Leonie and Wenk, Carola},
  title =	{{Realizability of Free Spaces of Curves}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{3:1--3:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.3},
  URN =		{urn:nbn:de:0030-drops-193057},
  doi =		{10.4230/LIPIcs.ISAAC.2023.3},
  annote =	{Keywords: Fr\'{e}chet distance, Distance Geometry, free space diagram, inverse problem}
}
Document
Short Paper
DOI: 10.4230/LIPIcs.GIScience.2023.18
Anonymous Routing Using Minimum Capacity Clustering (Short Paper)

Authors: Maike Buchin and Lukas Plätz

Published in: LIPIcs, Volume 277, 12th International Conference on Geographic Information Science (GIScience 2023)

Abstract
We present a framework which allows one to use an online routing service and get live updates without revealing the sensitive starting and ending points of one’s route. For that, we obfuscate the starting and ending locations in minimum capacity clusters and reveal only the route between these clusters. We compare different anonymous clustering strategies on positions in the network with efficient approximations and analyse the impact of the anonymisation on the route. We experimentally evaluate the effect of the anonymisation scheme in real-world settings.
Cite as

Maike Buchin and Lukas Plätz. Anonymous Routing Using Minimum Capacity Clustering (Short Paper). In 12th International Conference on Geographic Information Science (GIScience 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 277, pp. 18:1-18:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{buchin_et_al:LIPIcs.GIScience.2023.18,
  author =	{Buchin, Maike and Pl\"{a}tz, Lukas},
  title =	{{Anonymous Routing Using Minimum Capacity Clustering}},
  booktitle =	{12th International Conference on Geographic Information Science (GIScience 2023)},
  pages =	{18:1--18:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-288-4},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{277},
  editor =	{Beecham, Roger and Long, Jed A. and Smith, Dianna and Zhao, Qunshan and Wise, Sarah},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GIScience.2023.18},
  URN =		{urn:nbn:de:0030-drops-189131},
  doi =		{10.4230/LIPIcs.GIScience.2023.18},
  annote =	{Keywords: Anonymity, approximation Algorithms, directed Networks, minimum capacity Clustering, Privacy}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.29
Efficient Fréchet Distance Queries for Segments

Authors: Maike Buchin, Ivor van der Hoog, Tim Ophelders, Lena Schlipf, Rodrigo I. Silveira, and Frank Staals

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
We study the problem of constructing a data structure that can store a two-dimensional polygonal curve P, such that for any query segment ab one can efficiently compute the Fréchet distance between P and ab. First we present a data structure of size O(n log n) that can compute the Fréchet distance between P and a horizontal query segment ab in O(log n) time, where n is the number of vertices of P. In comparison to prior work, this significantly reduces the required space. We extend the type of queries allowed, as we allow a query to be a horizontal segment ab together with two points s, t ∈ P (not necessarily vertices), and ask for the Fréchet distance between ab and the curve of P in between s and t. Using O(nlog²n) storage, such queries take O(log³ n) time, simplifying and significantly improving previous results. We then generalize our results to query segments of arbitrary orientation. We present an O(nk^{3+ε}+n²) size data structure, where k ∈ [1,n] is a parameter the user can choose, and ε > 0 is an arbitrarily small constant, such that given any segment ab and two points s, t ∈ P we can compute the Fréchet distance between ab and the curve of P in between s and t in O((n/k)log²n+log⁴ n) time. This is the first result that allows efficient exact Fréchet distance queries for arbitrarily oriented segments. We also present two applications of our data structure. First, we show that our data structure allows us to compute a local δ-simplification (with respect to the Fréchet distance) of a polygonal curve in O(n^{5/2+ε}) time, improving a previous O(n³) time algorithm. Second, we show that we can efficiently find a translation of an arbitrary query segment ab that minimizes the Fréchet distance with respect to a subcurve of P.
Cite as

Maike Buchin, Ivor van der Hoog, Tim Ophelders, Lena Schlipf, Rodrigo I. Silveira, and Frank Staals. Efficient Fréchet Distance Queries for Segments. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{buchin_et_al:LIPIcs.ESA.2022.29,
  author =	{Buchin, Maike and van der Hoog, Ivor and Ophelders, Tim and Schlipf, Lena and Silveira, Rodrigo I. and Staals, Frank},
  title =	{{Efficient Fr\'{e}chet Distance Queries for Segments}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{29:1--29:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.29},
  URN =		{urn:nbn:de:0030-drops-169671},
  doi =		{10.4230/LIPIcs.ESA.2022.29},
  annote =	{Keywords: Computational Geometry, Data Structures, Fr\'{e}chet distance}
}
Document
DOI: 10.4230/DagRep.12.2.17
Computation and Reconfiguration in Low-Dimensional Topological Spaces (Dagstuhl Seminar 22062)

Authors: Maike Buchin, Anna Lubiw, Arnaud de Mesmay, Saul Schleimer, and Florestan Brunck

Published in: Dagstuhl Reports, Volume 12, Issue 2 (2022)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 22062 "Computation and Reconfiguration in Low-Dimensional Topological Spaces". The seminar consisted of a small collection of introductory talks, an open problem session, and then the seminar participants worked in small groups on problems on reconfiguration within the context of objects as diverse as elimination trees, morphings, curves on surfaces, translation surfaces and Delaunay triangulations.
Cite as

Maike Buchin, Anna Lubiw, Arnaud de Mesmay, Saul Schleimer, and Florestan Brunck. Computation and Reconfiguration in Low-Dimensional Topological Spaces (Dagstuhl Seminar 22062). In Dagstuhl Reports, Volume 12, Issue 2, pp. 17-66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{buchin_et_al:DagRep.12.2.17,
  author =	{Buchin, Maike and Lubiw, Anna and de Mesmay, Arnaud and Schleimer, Saul and Brunck, Florestan},
  title =	{{Computation and Reconfiguration in Low-Dimensional Topological Spaces (Dagstuhl Seminar 22062)}},
  pages =	{17--66},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2022},
  volume =	{12},
  number =	{2},
  editor =	{Buchin, Maike and Lubiw, Anna and de Mesmay, Arnaud and Schleimer, Saul and Brunck, Florestan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.2.17},
  URN =		{urn:nbn:de:0030-drops-169305},
  doi =		{10.4230/DagRep.12.2.17},
  annote =	{Keywords: Geometric Topology, Computational Geometry, Graph Drawing, Reconfiguration, Dagstuhl Seminar}
}
Document
DOI: 10.4230/LIPIcs.WABI.2020.13
Shape Decomposition Algorithms for Laser Capture Microdissection

Authors: Leonie Selbach, Tobias Kowalski, Klaus Gerwert, Maike Buchin, and Axel Mosig

Published in: LIPIcs, Volume 172, 20th International Workshop on Algorithms in Bioinformatics (WABI 2020)

Abstract
In the context of biomarker discovery and molecular characterization of diseases, laser capture microdissection is a highly effective approach to extract disease-specific regions from complex, heterogeneous tissue samples. These regions have to be decomposed into feasible fragments as they have to satisfy certain constraints in size and morphology for the extraction to be successful. We model this problem of constrained shape decomposition as the computation of optimal feasible decompositions of simple polygons. We use a skeleton-based approach and present an algorithmic framework that allows the implementation of various feasibility criteria as well as optimization goals. Motivated by our application, we consider different constraints and examine the resulting fragmentations. Furthermore, we apply our method to lung tissue samples and show its advantages in comparison to a heuristic decomposition approach.
Cite as

Leonie Selbach, Tobias Kowalski, Klaus Gerwert, Maike Buchin, and Axel Mosig. Shape Decomposition Algorithms for Laser Capture Microdissection. In 20th International Workshop on Algorithms in Bioinformatics (WABI 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 172, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{selbach_et_al:LIPIcs.WABI.2020.13,
  author =	{Selbach, Leonie and Kowalski, Tobias and Gerwert, Klaus and Buchin, Maike and Mosig, Axel},
  title =	{{Shape Decomposition Algorithms for Laser Capture Microdissection}},
  booktitle =	{20th International Workshop on Algorithms in Bioinformatics (WABI 2020)},
  pages =	{13:1--13:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-161-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{172},
  editor =	{Kingsford, Carl and Pisanti, Nadia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2020.13},
  URN =		{urn:nbn:de:0030-drops-128020},
  doi =		{10.4230/LIPIcs.WABI.2020.13},
  annote =	{Keywords: Laser capture microdissection, shape decomposition, skeletonization}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2019.50
The k-Fréchet Distance: How to Walk Your Dog While Teleporting

Authors: Hugo Alves Akitaya, Maike Buchin, Leonie Ryvkin, and Jérôme Urhausen

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Abstract
We introduce a new distance measure for comparing polygonal chains: the k-Fréchet distance. As the name implies, it is closely related to the well-studied Fréchet distance but detects similarities between curves that resemble each other only piecewise. The parameter k denotes the number of subcurves into which we divide the input curves (thus we allow up to k-1 "teleports" on each input curve). The k-Fréchet distance provides a nice transition between (weak) Fréchet distance and Hausdorff distance. However, we show that deciding this distance measure turns out to be NP-hard, which is interesting since both (weak) Fréchet and Hausdorff distance are computable in polynomial time. Nevertheless, we give several possibilities to deal with the hardness of the k-Fréchet distance: besides a short exponential-time algorithm for the general case, we give a polynomial-time algorithm for k=2, i.e., we ask that we subdivide our input curves into two subcurves each. We can also approximate the optimal k by factor 2. We then present a more intricate FPT algorithm using parameters k (the number of allowed subcurves) and z (the number of segments of one curve that intersect the epsilon-neighborhood of a point on the other curve).
Cite as

Hugo Alves Akitaya, Maike Buchin, Leonie Ryvkin, and Jérôme Urhausen. The k-Fréchet Distance: How to Walk Your Dog While Teleporting. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{alvesakitaya_et_al:LIPIcs.ISAAC.2019.50,
  author =	{Alves Akitaya, Hugo and Buchin, Maike and Ryvkin, Leonie and Urhausen, J\'{e}r\^{o}me},
  title =	{{The k-Fr\'{e}chet Distance: How to Walk Your Dog While Teleporting}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{50:1--50:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.50},
  URN =		{urn:nbn:de:0030-drops-115462},
  doi =		{10.4230/LIPIcs.ISAAC.2019.50},
  annote =	{Keywords: Measures, Fr\'{e}chet distance, Hardness, FPT}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2019.55
Distance Measures for Embedded Graphs

Authors: Hugo A. Akitaya, Maike Buchin, Bernhard Kilgus, Stef Sijben, and Carola Wenk

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Abstract
We introduce new distance measures for comparing straight-line embedded graphs based on the Fréchet distance and the weak Fréchet distance. These graph distances are defined using continuous mappings and thus take the combinatorial structure as well as the geometric embeddings of the graphs into account. We present a general algorithmic approach for computing these graph distances. Although we show that deciding the distances is NP-hard for general embedded graphs, we prove that our approach yields polynomial time algorithms if the graphs are trees, and for the distance based on the weak Fréchet distance if the graphs are planar embedded. Moreover, we prove that deciding the distances based on the Fréchet distance remains NP-hard for planar embedded graphs and show how our general algorithmic approach yields an exponential time algorithm and a polynomial time approximation algorithm for this case. Our work combines and extends the work of Buchin et al. [Maike Buchin et al., 2017] and Akitaya et al. [Hugo Akitaya et al., 2019] presented at EuroCG.
Cite as

Hugo A. Akitaya, Maike Buchin, Bernhard Kilgus, Stef Sijben, and Carola Wenk. Distance Measures for Embedded Graphs. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{akitaya_et_al:LIPIcs.ISAAC.2019.55,
  author =	{Akitaya, Hugo A. and Buchin, Maike and Kilgus, Bernhard and Sijben, Stef and Wenk, Carola},
  title =	{{Distance Measures for Embedded Graphs}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{55:1--55:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.55},
  URN =		{urn:nbn:de:0030-drops-115517},
  doi =		{10.4230/LIPIcs.ISAAC.2019.55},
  annote =	{Keywords: Fr\'{e}chet distance, graph comparison, embedded graphs}
}
Document
DOI: 10.4230/DagRep.4.3.138
Interaction and Collective Movement Processing (Dagstuhl Seminar 14132)

Authors: Maike Buchin, Luca Giuggioli, van Kreveld. Marc, and Guy Theraulaz

Published in: Dagstuhl Reports, Volume 4, Issue 3 (2014)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 14132 ``Interaction and Collective Movement Processing''. This seminar brought together a group of 30 scientists with varied backgrounds, but with a shared interest in computations involved in the processing of moving entity data, like humans or animals. The seminar focused on characterizing and modelling interaction between moving entities, and featured four invited talks in four main research fields: ecology, computational geometry, GIScience, and collective motion. The remainder of the program consisted of short presentations, open problem sessions, break-out groups to work on open problems, and reporting sessions based on research done in the break-out groups.
Cite as

Maike Buchin, Luca Giuggioli, van Kreveld. Marc, and Guy Theraulaz. Interaction and Collective Movement Processing (Dagstuhl Seminar 14132). In Dagstuhl Reports, Volume 4, Issue 3, pp. 138-152, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@Article{buchin_et_al:DagRep.4.3.138,
  author =	{Buchin, Maike and Giuggioli, Luca and van Kreveld. Marc and Theraulaz, Guy},
  title =	{{Interaction and Collective Movement Processing (Dagstuhl Seminar 14132)}},
  pages =	{138--152},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2014},
  volume =	{4},
  number =	{3},
  editor =	{Buchin, Maike and Giuggioli, Luca and van Kreveld. Marc and Theraulaz, Guy},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.3.138},
  URN =		{urn:nbn:de:0030-drops-45953},
  doi =		{10.4230/DagRep.4.3.138},
  annote =	{Keywords: collective movement, moving entity, computational geometry, GIScience, ecology, collective motion, trajectory, movement analysis}
}
Document
DOI: 10.4230/DagSemProc.10491.3
10491 Results of the break-out group: Aggregation

Authors: Mark de Berg, Jörg-Rüdiger Sack, Bettina Speckmann, Anne Driemel, Maike Buchin, Monika Sester, and Marc van Kreveld

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

Abstract
We discussed different problems that arise when aggregating trajectories: how to segment the input, whether to use original parts of the input trajectories, as opposed to an ``averaged'' path and how to simplify the aggregated structure. We give examples where these questions are not easily answered.
Cite as

Mark de Berg, Jörg-Rüdiger Sack, Bettina Speckmann, Anne Driemel, Maike Buchin, Monika Sester, and Marc van Kreveld. 10491 Results of the break-out group: Aggregation. In Representation, Analysis and Visualization of Moving Objects. Dagstuhl Seminar Proceedings, Volume 10491, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{deberg_et_al:DagSemProc.10491.3,
  author =	{de Berg, Mark and Sack, J\"{o}rg-R\"{u}diger and Speckmann, Bettina and Driemel, Anne and Buchin, Maike and Sester, Monika and van Kreveld, Marc},
  title =	{{10491 Results of the break-out group: Aggregation}},
  booktitle =	{Representation, Analysis and Visualization of Moving Objects},
  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.10491.3},
  URN =		{urn:nbn:de:0030-drops-29878},
  doi =		{10.4230/DagSemProc.10491.3},
  annote =	{Keywords: Aggregation, Trajectories, Generalization, Map Generation}
}
Document
DOI: 10.4230/DagSemProc.10491.5
10491 Results of the break-out group: Gulls Data

Authors: Emiel van Loon, Jörg-Rüdiger Sack, Kevin Buchin, Maike Buchin, Mark de Berg, Marc van Kreveld, Joachim Gudmundsson, and David Mountain

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

Abstract
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.
Cite as

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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