On Optimal Polyline Simplification using the Hausdorff and Fréchet Distance

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Abstract

We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given epsilon>0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most epsilon. We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the algorithms are given a considerably larger error threshold than epsilon. Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard. The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time. Finally, to compute the optimal simplification from a polygonal line consisting of n vertices under the Fréchet distance, we give an O(kn^5) time algorithm that requires O(kn^2) space, where k is the output complexity of the simplification.
Original languageEnglish
Title of host publication34th International Symposium on Computational Geometry
Subtitle of host publicationSoCG 2018, June 11–14, 2018, Budapest, Hungary
EditorsBettina Speckmann, Csaba D. Tóth
PublisherLeibniz International Proceedings in Informatics (LIPIcs)
Pages56:1-56:14
Number of pages14
ISBN (Print)978-3-95977-066-8
DOIs
Publication statusPublished - 8 Jun 2018

Publication series

NameLeibniz International Proceedings in Informatics
Volume99

Keywords

  • polygonal line simplification
  • Hausdorff distance
  • Fréchet distance
  • Imai-Iri
  • Douglas-Peucker

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