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

Authors Hugo Alves Akitaya, Maike Buchin, Leonie Ryvkin, Jérôme Urhausen

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Fixed parameter tractability
Keywords
  • Measures
  • Fréchet distance
  • Hardness
  • FPT

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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).

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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) https://doi.org/10.4230/LIPIcs.ISAAC.2019.50

Author Details

Hugo Alves Akitaya
  • Department of Computer Science, Tufts University, Massachusetts, USA
Maike Buchin
  • Department of Mathematics, Ruhr University Bochum, Germany
Leonie Ryvkin
  • Department of Mathematics, Ruhr University Bochum, Germany
Jérôme Urhausen
  • Department of Information and Computing Sciences, Utrecht University, Netherlands

Funding

  • Alves Akitaya, Hugo: supported by NSF awards CCF-1422311 and CCF-1423615, and the Science Without Borders scholarship program.
  • Urhausen, Jérôme: supported by the Netherlands Organisation for Scientific Research under project 612.001.651.

Acknowledgements

We would like to thank Erik Demaine for contributing the key idea for proving hardness in the free space diagram in Section 3.1, as well as the organizers and other participants of the Intensive Research Program in Discrete, Combinatorial and Computational Geometry in Barcelona, 2018, for providing the perfect environment to meet other researchers.

References

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