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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.

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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}
}

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DOI: 10.4230/LIPIcs.SoCG.2023.27

Computing a Dirichlet Domain for a Hyperbolic Surface

Authors: Vincent Despré, Benedikt Kolbe, Hugo Parlier, and Monique Teillaud

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract

This paper exhibits and analyzes an algorithm that takes a given closed orientable hyperbolic surface and outputs an explicit Dirichlet domain. The input is a fundamental polygon with side pairings. While grounded in topological considerations, the algorithm makes key use of the geometry of the surface. We introduce data structures that reflect this interplay between geometry and topology and show that the algorithm runs in polynomial time, in terms of the initial perimeter and the genus of the surface.

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Vincent Despré, Benedikt Kolbe, Hugo Parlier, and Monique Teillaud. Computing a Dirichlet Domain for a Hyperbolic Surface. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{despre_et_al:LIPIcs.SoCG.2023.27,
  author =	{Despr\'{e}, Vincent and Kolbe, Benedikt and Parlier, Hugo and Teillaud, Monique},
  title =	{{Computing a Dirichlet Domain for a Hyperbolic Surface}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{27:1--27:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.27},
  URN =		{urn:nbn:de:0030-drops-178771},
  doi =		{10.4230/LIPIcs.SoCG.2023.27},
  annote =	{Keywords: Hyperbolic geometry, Topology, Voronoi diagram, Algorithm}
}

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Documents authored by Kolbe, Benedikt

Fast Approximations and Coresets for (k,𝓁)-Median Under Dynamic Time Warping

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Computing a Dirichlet Domain for a Hyperbolic Surface

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