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Computing the Fréchet Distance Between Uncertain Curves in One Dimension

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Algorithms and Data Structures (WADS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12808))

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Abstract

We consider the problem of computing the Fréchet distance between two curves for which the exact locations of the vertices are unknown. Each vertex may be placed in a given uncertainty region for that vertex, and the objective is to place vertices so as to minimise the Fréchet distance. This problem was recently shown to be NP-hard in 2D, and it is unclear how to compute an optimal vertex placement at all.

We give a polynomial-time algorithm for 1D curves with intervals as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D in the case that vertices are placed to maximise the Fréchet distance.

We also study the weak Fréchet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than for the regular Fréchet distance—and indeed we can easily prove that the problem is NP-hard in 2D—the optimal placement of vertices in 1D can be computed in polynomial time. Finally, we investigate the discrete weak Fréchet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D.

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Acknowledgements

Research on the topic of this paper was initiated at the 5th Workshop on Applied Geometric Algorithms (AGA 2020) in Langbroek, Netherlands. Maarten Löffler is partially supported by the Dutch Research Council (NWO) under project no. 614.001.504 and no. 628.011.005. Aleksandr Popov is supported by the Dutch Research Council (NWO) under project no. 612.001.801. Jérôme Urhausen is supported by the Dutch Research Council (NWO) under project no. 612.001.651.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, TU Eindhoven, Eindhoven, The Netherlands

    Kevin Buchin, Tim Ophelders, Aleksandr Popov & Kevin Verbeek

  2. Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands

    Maarten Löffler, Tim Ophelders & Jérôme Urhausen

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  1. Kevin Buchin
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Correspondence to Aleksandr Popov .

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Editors and Affiliations

  1. University of Waterloo, Waterloo, ON, Canada

    Anna Lubiw

  2. University of Alberta, Edmonton, AB, Canada

    Mohammad Salavatipour

  3. Dalhousie University, Halifax, Canada

    Meng He

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Buchin, K., Löffler, M., Ophelders, T., Popov, A., Urhausen, J., Verbeek, K. (2021). Computing the Fréchet Distance Between Uncertain Curves in One Dimension. In: Lubiw, A., Salavatipour, M., He, M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science(), vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_18

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