11 Search Results for "Conradi, Jacobus"

Document
DOI: 10.4230/LIPIcs.ESA.2025.64
A Dynamic Piecewise-Linear Geometric Index with Worst-Case Guarantees

Authors: Emil Toftegaard Gæde, Ivor van der Hoog, Eva Rotenberg, and Tord Stordalen

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

Abstract
Indexing data is a fundamental problem in computer science. The input is a set S of n distinct integers from a universe 𝒰. Indexing queries take a value q ∈ 𝒰 and return the membership, predecessor or rank of q in S. A range query takes two values q, r ∈ 𝒰 and returns the set S ∩ [q,r]. Recently, various papers study a special case where the the input data behaves in an approximately piece-wise linear way. Given the sorted (rank,value) pairs, and given some constant ε, one wants to maintain a small number of axis-disjoint line-segments such that, for each rank, the value is within ± ε of the corresponding line-segment. Ferragina and Vinciguerra (VLDB 2020) observe that this geometric problem is useful for solving indexing problems, particularly when the number of line-segments is small compared to the size of the dataset. We study the dynamic version of this geometric problem. In the dynamic setting, inserting or deleting just one data point may cause up to three line-segments to be merged, or one line-segment to be split at most three-way. To determine and compute this, we use techniques from dynamic maintenance of convex hulls, and provide new algorithms with worst-case guarantees, including an O(log n) algorithm to compute a separating line between two non-intersecting convex hulls - an operation previously missing from the literature. We then use our fully-dynamic geometry-based subroutine in an indexing data structure, combining it with a natural hashing technique. The resulting indexing data structure has theoretically efficient worst-case guarantees in expectation. We compare its practical performance to the solution of Ferragina and Vinciguerra, which was shown to perform better in certain structured settings [Sun, Zhou, Li VLDB 2023]. Our empirical analysis shows that our solution supports more efficient range queries in the special case where the update sequence contains many deletions.
Cite as

Emil Toftegaard Gæde, Ivor van der Hoog, Eva Rotenberg, and Tord Stordalen. A Dynamic Piecewise-Linear Geometric Index with Worst-Case Guarantees. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 64:1-64:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gaede_et_al:LIPIcs.ESA.2025.64,
  author =	{G{\ae}de, Emil Toftegaard and van der Hoog, Ivor and Rotenberg, Eva and Stordalen, Tord},
  title =	{{A Dynamic Piecewise-Linear Geometric Index with Worst-Case Guarantees}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{64:1--64:18},
  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.2025.64},
  URN =		{urn:nbn:de:0030-drops-245323},
  doi =		{10.4230/LIPIcs.ESA.2025.64},
  annote =	{Keywords: Algorithms Engineering, Data Structures, Indexing, Convex Hulls}
}
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}
}
Artifact
Software
DOI: 10.4230/artifacts.23288
nonObtuseTri

Authors: Mikkel Abrahamsen, Florestan Brunck, Jacobus Conradi, Benedikt Kolbe, and André Nusser

Abstract
Cite as

Mikkel Abrahamsen, Florestan Brunck, Jacobus Conradi, Benedikt Kolbe, André Nusser. nonObtuseTri (Software, Source Code). Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@misc{dagstuhl-artifact-23288,
   title = {{nonObtuseTri}}, 
   author = {Abrahamsen, Mikkel and Brunck, Florestan and Conradi, Jacobus and Kolbe, Benedikt and Nusser, Andr\'{e}},
   note = {Software, version v3.0., swhId: \href{https://archive.softwareheritage.org/swh:1:dir:b6301970ab626c084f309f9b5e6b91e9acf24949;origin=https://github.com/JacobusTheSecond/nonObtuseTri;visit=swh:1:snp:4b7f079e3cd897cb0826392670078921ce262ba2;anchor=swh:1:rev:bd1842a4272b4e01ad623cf6bb02c7617c3da98a}{\texttt{swh:1:dir:b6301970ab626c084f309f9b5e6b91e9acf24949}} (visited on 2025-06-20)},
   url = {https://github.com/JacobusTheSecond/nonObtuseTri},
   doi = {10.4230/artifacts.23288},
}
Document
CG Challenge
DOI: 10.4230/LIPIcs.SoCG.2025.79
Computing Non-Obtuse Triangulations with Few Steiner Points (CG Challenge)

Authors: Mikkel Abrahamsen, Florestan Brunck, Jacobus Conradi, Benedikt Kolbe, and André Nusser

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

Abstract
We present the winning implementation of the Seventh Computational Geometry Challenge (CG:SHOP 2025). The task in this challenge was to find non-obtuse triangulations for given planar regions, respecting a given set of constraints consisting of extra vertices and edges that must be part of the triangulation. The goal was to minimize the number of introduced Steiner points. Our approach is to maintain a constrained Delaunay triangulation, for which we repeatedly remove, relocate, or add Steiner points. We use local search to choose the action that improves the triangulation the most, until the resulting triangulation is non-obtuse.
Cite as

Mikkel Abrahamsen, Florestan Brunck, Jacobus Conradi, Benedikt Kolbe, and André Nusser. Computing Non-Obtuse Triangulations with Few Steiner Points (CG Challenge). In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 79:1-79:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2025.79,
  author =	{Abrahamsen, Mikkel and Brunck, Florestan and Conradi, Jacobus and Kolbe, Benedikt and Nusser, Andr\'{e}},
  title =	{{Computing Non-Obtuse Triangulations with Few Steiner Points}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{79:1--79:13},
  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.2025.79},
  URN =		{urn:nbn:de:0030-drops-232311},
  doi =		{10.4230/LIPIcs.SoCG.2025.79},
  annote =	{Keywords: non-obtuse triangulation, local search, competition}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.22
Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D

Authors: Lotte Blank, Jacobus Conradi, Anne Driemel, Benedikt Kolbe, André Nusser, and Marena Richter

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

Abstract
The Fréchet distance is a computational mainstay for comparing polygonal curves. The Fréchet distance under translation, which is a translation invariant version, considers the similarity of two curves independent of their location in space. It is defined as the minimum Fréchet distance that arises from allowing arbitrary translations of the input curves. This problem and numerous variants of the Fréchet distance under some transformations have been studied, with more work concentrating on the discrete Fréchet distance, leaving a significant gap between the discrete and continuous versions of the Fréchet distance under transformations. Our contribution is twofold: First, we present an algorithm for the Fréchet distance under translation on 1-dimensional curves of complexity n with a running time of 𝒪(n^{8/3} log³ n). To achieve this, we develop a novel framework for the problem for 1-dimensional curves, which also applies to other scenarios and leads to our second contribution. We present an algorithm with the same running time of 𝒪(n^{8/3} log³ n) for the Fréchet distance under scaling for 1-dimensional curves. For both algorithms we match the running times of the discrete case and improve the previously best known bounds of 𝒪̃(n⁴). Our algorithms rely on technical insights but are conceptually simple, essentially reducing the continuous problem to the discrete case across different length scales.
Cite as

Lotte Blank, Jacobus Conradi, Anne Driemel, Benedikt Kolbe, André Nusser, and Marena Richter. Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blank_et_al:LIPIcs.SoCG.2025.22,
  author =	{Blank, Lotte and Conradi, Jacobus and Driemel, Anne and Kolbe, Benedikt and Nusser, Andr\'{e} and Richter, Marena},
  title =	{{Transforming Dogs on the Line: On the Fr\'{e}chet Distance Under Translation or Scaling in 1D}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{22:1--22:16},
  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.2025.22},
  URN =		{urn:nbn:de:0030-drops-231746},
  doi =		{10.4230/LIPIcs.SoCG.2025.22},
  annote =	{Keywords: Fr\'{e}chet distance under translation, Fr\'{e}chet distance under scaling, time series, 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.SoCG.2024.42
Fast Approximations and Coresets for (k,𝓁)-Median Under Dynamic Time Warping

Authors: Jacobus Conradi, Benedikt Kolbe, Ioannis Psarros, and Dennis Rohde

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

Abstract
We present algorithms for the computation of ε-coresets for k-median clustering of point sequences in ℝ^d under the p-dynamic time warping (DTW) distance. Coresets under DTW have not been investigated before, and the analysis is not directly accessible to existing methods as DTW is not a metric. The three main ingredients that allow our construction of coresets are the adaptation of the ε-coreset framework of sensitivity sampling, bounds on the VC dimension of approximations to the range spaces of balls under DTW, and new approximation algorithms for the k-median problem under DTW. We achieve our results by investigating approximations of DTW that provide a trade-off between the provided accuracy and amenability to known techniques. In particular, we observe that given n curves under DTW, one can directly construct a metric that approximates DTW on this set, permitting the use of the wealth of results on metric spaces for clustering purposes. The resulting approximations are the first with polynomial running time and achieve a very similar approximation factor as state-of-the-art techniques. We apply our results to produce a practical algorithm approximating (k,𝓁)-median clustering under DTW.
Cite as

Jacobus Conradi, Benedikt Kolbe, Ioannis Psarros, and Dennis Rohde. Fast Approximations and Coresets for (k,𝓁)-Median Under Dynamic Time Warping. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{conradi_et_al:LIPIcs.SoCG.2024.42,
  author =	{Conradi, Jacobus and Kolbe, Benedikt and Psarros, Ioannis and Rohde, Dennis},
  title =	{{Fast Approximations and Coresets for (k,𝓁)-Median Under Dynamic Time Warping}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{42:1--42:17},
  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.2024.42},
  URN =		{urn:nbn:de:0030-drops-199875},
  doi =		{10.4230/LIPIcs.SoCG.2024.42},
  annote =	{Keywords: Dynamic time warping, coreset, median clustering, approximation algorithm}
}
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}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.46
On Computing the k-Shortcut Fréchet Distance

Authors: Jacobus Conradi and Anne Driemel

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
The Fréchet distance is a popular measure of dissimilarity for polygonal curves. It is defined as a min-max formulation that considers all direction-preserving continuous bijections of the two curves. Because of its susceptibility to noise, Driemel and Har-Peled introduced the shortcut Fréchet distance in 2012, where one is allowed to take shortcuts along one of the curves, similar to the edit distance for sequences. We analyse the parameterized version of this problem, where the number of shortcuts is bounded by a parameter k. The corresponding decision problem can be stated as follows: Given two polygonal curves T and B of at most n vertices, a parameter k and a distance threshold δ, is it possible to introduce k shortcuts along B such that the Fréchet distance of the resulting curve and the curve T is at most δ? We study this problem for polygonal curves in the plane. We provide a complexity analysis for this problem with the following results: (i) assuming the exponential-time-hypothesis (ETH), there exists no algorithm with running time bounded by n^o(k); (ii) there exists a decision algorithm with running time in O(k n^{2k+2} log n). In contrast, we also show that efficient approximate decider algorithms are possible, even when k is large. We present a (3+ε)-approximate decider algorithm with running time in O(k n² log² n) for fixed ε. In addition, we can show that, if k is a constant and the two curves are c-packed for some constant c, then the approximate decider algorithm runs in near-linear time.
Cite as

Jacobus Conradi and Anne Driemel. On Computing the k-Shortcut Fréchet Distance. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 46:1-46:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{conradi_et_al:LIPIcs.ICALP.2022.46,
  author =	{Conradi, Jacobus and Driemel, Anne},
  title =	{{On Computing the k-Shortcut Fr\'{e}chet Distance}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{46:1--46:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.46},
  URN =		{urn:nbn:de:0030-drops-163875},
  doi =		{10.4230/LIPIcs.ICALP.2022.46},
  annote =	{Keywords: Fr\'{e}chet distance, Partial similarity, Conditional lower bounds, Approximation algorithms}
}
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