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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.

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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}
}

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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}
}

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Realizability of Free Spaces of Curves

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The k-Fréchet Distance: How to Walk Your Dog While Teleporting

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