5 Search Results for "Brüning, Frederik"

Document
DOI: 10.4230/LIPIcs.ESA.2025.12
Subtrajectory Clustering and Coverage Maximization in Cubic Time, or Better

Authors: Jacobus Conradi and Anne Driemel

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
Many application areas collect unstructured trajectory data. In subtrajectory clustering, one is interested to find patterns in this data using a hybrid combination of segmentation and clustering. We analyze two variants of this problem based on the well-known SetCover and CoverageMaximization problems. In both variants the set system is induced by metric balls under the Fréchet distance centered at polygonal curves. Our algorithms focus on improving the running time of the update step of the generic greedy algorithm by means of a careful combination of sweeps through a candidate space. In the first variant, we are given a polygonal curve P of complexity n, distance threshold Δ and complexity bound 𝓁 and the goal is to identify a minimum-size set of center curves 𝒞, where each center curve is of complexity at most 𝓁 and every point p on P is covered. A point p on P is covered if it is part of a subtrajectory π_p of P such that there is a center c ∈ 𝒞 whose Fréchet distance to π_p is at most Δ. We present an approximation algorithm for this problem with a running time of 𝒪((n²𝓁 + √{k_Δ}n^{5/2})log²n), where k_Δ is the size of an optimal solution. The algorithm gives a bicriterial approximation guarantee that relaxes the Fréchet distance threshold by a constant factor and the size of the solution by a factor of 𝒪(log n). The second problem variant asks for the maximum fraction of the input curve P that can be covered using k center curves, where k ≤ n is a parameter to the algorithm. For the second problem variant, our techniques lead to an algorithm with a running time of 𝒪((k+𝓁)n²log²n) and similar approximation guarantees. Note that in both algorithms k,k_Δ ∈ O(n) and hence the running time is cubic, or better if k ≪ n.
Cite as

Jacobus Conradi and Anne Driemel. Subtrajectory Clustering and Coverage Maximization in Cubic Time, or Better. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{conradi_et_al:LIPIcs.ESA.2025.12,
  author =	{Conradi, Jacobus and Driemel, Anne},
  title =	{{Subtrajectory Clustering and Coverage Maximization in Cubic Time, or Better}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{12:1--12:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.12},
  URN =		{urn:nbn:de:0030-drops-244806},
  doi =		{10.4230/LIPIcs.ESA.2025.12},
  annote =	{Keywords: Clustering, Set cover, Fr\'{e}chet distance, Approximation algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.133
Faster, Deterministic and Space Efficient Subtrajectory Clustering

Authors: Ivor van der Hoog, Thijs van der Horst, and Tim Ophelders

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Given a trajectory T and a distance Δ, we wish to find a set C of curves of complexity at most 𝓁, such that we can cover T with subcurves that each are within Fréchet distance Δ to at least one curve in C. We call C an (𝓁,Δ)-clustering and aim to find an (𝓁,Δ)-clustering of minimum cardinality. This problem variant was introduced by Akitaya et al. (2021) and shown to be NP-complete. The main focus has therefore been on bicriteria approximation algorithms, allowing for the clustering to be an (𝓁, Θ(Δ))-clustering of roughly optimal size. We present algorithms that construct (𝓁,4Δ)-clusterings of 𝒪(k log n) size, where k is the size of the optimal (𝓁, Δ)-clustering. We use 𝒪(n³) space and 𝒪(k n³ log⁴ n) time. Our algorithms significantly improve upon the clustering quality (improving the approximation factor in Δ) and size (whenever 𝓁 ∈ Ω(log n / log k)). We offer deterministic running times improving known expected bounds by a factor near-linear in 𝓁. Additionally, we match the space usage of prior work, and improve it substantially, by a factor super-linear in n𝓁, when compared to deterministic results.
Cite as

Ivor van der Hoog, Thijs van der Horst, and Tim Ophelders. Faster, Deterministic and Space Efficient Subtrajectory Clustering. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 133:1-133:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanderhoog_et_al:LIPIcs.ICALP.2025.133,
  author =	{van der Hoog, Ivor and van der Horst, Thijs and Ophelders, Tim},
  title =	{{Faster, Deterministic and Space Efficient Subtrajectory Clustering}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{133:1--133:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.133},
  URN =		{urn:nbn:de:0030-drops-235109},
  doi =		{10.4230/LIPIcs.ICALP.2025.133},
  annote =	{Keywords: Fr\'{e}chet distance, clustering, set cover}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.36
Faster Fréchet Distance Under Transformations

Authors: Kevin Buchin, Maike Buchin, Zijin Huang, André Nusser, and Sampson Wong

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study the problem of computing the Fréchet distance between two polygonal curves under transformations. First, we consider translations in the Euclidean plane. Given two curves π and σ of total complexity n and a threshold δ ≥ 0, we present an 𝒪̃(n^{7 + 1/3}) time algorithm to determine whether there exists a translation t ∈ ℝ² such that the Fréchet distance between π and σ + t is at most δ. This improves on the previous best result, which is an 𝒪(n⁸) time algorithm. We then generalize this result to any class of rationally parameterized transformations, which includes translation, rotation, scaling, and arbitrary affine transformations. For a class T of rationally parametrized transformations with k degrees of freedom, we show that one can determine whether there is a transformation τ ∈ T such that the Fréchet distance between π and τ(σ) is at most δ in 𝒪̃(n^{3k+4/3}) time.
Cite as

Kevin Buchin, Maike Buchin, Zijin Huang, André Nusser, and Sampson Wong. Faster Fréchet Distance Under Transformations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{buchin_et_al:LIPIcs.ICALP.2025.36,
  author =	{Buchin, Kevin and Buchin, Maike and Huang, Zijin and Nusser, Andr\'{e} and Wong, Sampson},
  title =	{{Faster Fr\'{e}chet Distance Under Transformations}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{36:1--36:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.36},
  URN =		{urn:nbn:de:0030-drops-234137},
  doi =		{10.4230/LIPIcs.ICALP.2025.36},
  annote =	{Keywords: Fr\'{e}chet distance, curve similarity, shape matching}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.78
Efficient Greedy Discrete Subtrajectory Clustering

Authors: Ivor van der Hoog, Lara Ost, Eva Rotenberg, and Daniel Rutschmann

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We cluster a set of trajectories 𝒯 using subtrajectories of 𝒯. We require for a clustering C that any two subtrajectories (𝒯[a, b], 𝒯[c, d]) in a cluster have disjoint intervals [a,b] and [c, d]. Clustering quality may be measured by the number of clusters, the number of vertices of 𝒯 that are absent from the clustering, and by the Fréchet distance between subtrajectories in a cluster. A Δ-cluster of 𝒯 is a cluster 𝒫 of subtrajectories of 𝒯 with a centre P ∈ 𝒫, where all subtrajectories in 𝒫 have Fréchet distance at most Δ to P. Buchin, Buchin, Gudmundsson, Löffler and Luo present two O(n² + n m 𝓁)-time algorithms: SC(max, 𝓁, Δ, 𝒯) computes a single Δ-cluster where P has at least 𝓁 vertices and maximises the cardinality m of 𝒫. SC(m, max, Δ, 𝒯) computes a single Δ-cluster where 𝒫 has cardinality m and maximises the complexity 𝓁 of P. In this paper, which is a mixture of algorithms engineering and theoretical insights, we use such maximum-cardinality clusters in a greedy clustering algorithm. We first provide an efficient implementation of SC(max, 𝓁, Δ, 𝒯) and SC(m, max, Δ, 𝒯) that significantly outperforms previous implementations. Next, we use these functions as a subroutine in a greedy clustering algorithm, which performs well when compared to existing subtrajectory clustering algorithms on real-world data. Finally, we observe that, for fixed Δ and 𝒯, these two functions always output a point on the Pareto front of some bivariate function θ(𝓁, m). We design a new algorithm PSC(Δ, 𝒯) that in O(n² log⁴ n) time computes a 2-approximation of this Pareto front. This yields a broader set of candidate clusters, with comparable quality to the output of the previous functions. We show that using PSC(Δ, 𝒯) as a subroutine improves the clustering quality and performance even further.
Cite as

Ivor van der Hoog, Lara Ost, Eva Rotenberg, and Daniel Rutschmann. Efficient Greedy Discrete Subtrajectory Clustering. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 78:1-78:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{vanderhoog_et_al:LIPIcs.SoCG.2025.78,
  author =	{van der Hoog, Ivor and Ost, Lara and Rotenberg, Eva and Rutschmann, Daniel},
  title =	{{Efficient Greedy Discrete Subtrajectory Clustering}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{78:1--78:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.78},
  URN =		{urn:nbn:de:0030-drops-232308},
  doi =		{10.4230/LIPIcs.SoCG.2025.78},
  annote =	{Keywords: Algorithms engineering, Fr\'{e}chet distance, subtrajectory clustering}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.28
Faster Approximate Covering of Subcurves Under the Fréchet Distance

Authors: Frederik Brüning, Jacobus Conradi, and Anne Driemel

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

Abstract
Subtrajectory clustering is an important variant of the trajectory clustering problem, where the start and endpoints of trajectory patterns within the collected trajectory data are not known in advance. We study this problem in the form of a set cover problem for a given polygonal curve: find the smallest number k of representative curves such that any point on the input curve is contained in a subcurve that has Fréchet distance at most a given Δ to a representative curve. We focus on the case where the representative curves are line segments and approach this NP-hard problem with classical techniques from the area of geometric set cover: we use a variant of the multiplicative weights update method which was first suggested by Brönniman and Goodrich for set cover instances with small VC-dimension. We obtain a bicriteria-approximation algorithm that computes a set of O(klog(k)) line segments that cover a given polygonal curve of n vertices under Fréchet distance at most O(Δ). We show that the algorithm runs in Õ(k² n + k n³) time in expectation and uses Õ(k n + n³) space. For input curves that are c-packed and lie in the plane, we bound the expected running time by Õ(k² c² n) and the space by Õ(kn + c² n). In addition, we present a variant of the algorithm that uses implicit weight updates on the candidate set and thereby achieves near-linear running time in n without any assumptions on the input curve, while keeping the same approximation bounds. This comes at the expense of a small (polylogarithmic) dependency on the relative arclength.
Cite as

Frederik Brüning, Jacobus Conradi, and Anne Driemel. Faster Approximate Covering of Subcurves Under the Fréchet Distance. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bruning_et_al:LIPIcs.ESA.2022.28,
  author =	{Br\"{u}ning, Frederik and Conradi, Jacobus and Driemel, Anne},
  title =	{{Faster Approximate Covering of Subcurves Under the Fr\'{e}chet Distance}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{28:1--28:16},
  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.2022.28},
  URN =		{urn:nbn:de:0030-drops-169660},
  doi =		{10.4230/LIPIcs.ESA.2022.28},
  annote =	{Keywords: Clustering, Set cover, Fr\'{e}chet distance, Approximation algorithms}
}
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