Document

# On Optimal Polyline Simplification Using the Hausdorff and Fréchet Distance

## File

LIPIcs.SoCG.2018.56.pdf
• Filesize: 0.78 MB
• 14 pages

## Cite As

Marc van Kreveld, Maarten Löffler, and Lionov Wiratma. On Optimal Polyline Simplification Using the Hausdorff and Fréchet Distance. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.56

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

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Mohammad Ali Abam, Mark de Berg, Peter Hachenberger, and Alireza Zarei:. Streaming algorithms for line simplification. Discrete & Computational Geometry, 43(3):497-515, 2010.
2. Pankaj K. Agarwal, Sariel Har-Peled, Nabil H. Mustafa, and Yusu Wang. Near-linear time approximation algorithms for curve simplification. Algorithmica, 42(3):203-219, 2005.
3. Helmut Alt, Bernd Behrends, and Johannes Blömer. Approximate matching of polygonal shapes. Annals of Mathematics and Artificial Intelligence, 13(3):251-265, Sep 1995.
4. Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry & Applications, 5(1-2):75-91, 1995.
5. Gill Barequet, Danny Z. Chen, Ovidiu Daescu, Michael T. Goodrich, and Jack Snoeyink. Efficiently approximating polygonal paths in three and higher dimensions. Algorithmica, 33(2):150-167, 2002.
6. Lilian Buzer. Optimal simplification of polygonal chain for rendering. In Proceedings 23rd Annual ACM Symposium on Computational Geometry, SCG '07, pages 168-174, 2007.
7. Jérémie Chalopin and Daniel Gonçalves. Every planar graph is the intersection graph of segments in the plane: Extended abstract. In Proceedings 41st Annual ACM Symposium on Theory of Computing, STOC '09, pages 631-638, 2009.
8. W.S. Chan and F. Chin. Approximation of polygonal curves with minimum number of line segments or minimum error. International Journal of Computational Geometry & Applications, 06(01):59-77, 1996.
9. Danny Z. Chen, Ovidiu Daescu, John Hershberger, Peter M. Kogge, Ningfang Mi, and Jack Snoeyink. Polygonal path simplification with angle constraints. Computational Geometry, 32(3):173-187, 2005.
10. Mark de Berg, Marc van Kreveld, and Stefan Schirra. Topologically correct subdivision simplification using the bandwidth criterion. Cartography and Geographic Information Systems, 25(4):243-257, 1998.
11. David H. Douglas and Thomas K. Peucker. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Cartographica, 10(2):112-122, 1973.
12. Regina Estkowski and Joseph S. B. Mitchell. Simplifying a polygonal subdivision while keeping it simple. In Proceedings 17th Annual ACM Symposium on Computational Geometry, SCG '01, pages 40-49, 2001.
13. Stefan Funke, Thomas Mendel, Alexander Miller, Sabine Storandt, and Maria Wiebe. Map simplification with topology constraints: Exactly and in practice. In Proc. 19th Workshop on Algorithm Engineering and Experiments (ALENEX), pages 185-196, 2017.
14. M.R. Garey, D.S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1(3):237-267, 1976.
15. Michael Godau. A natural metric for curves - computing the distance for polygonal chains and approximation algorithms. In Proceedings 8th Annual Symposium on Theoretical Aspects of Computer Science, STACS 91, pages 127-136. Springer-Verlag, 1991.
16. Leonidas J. Guibas, John E. Hershberger, Joseph S.B. Mitchell, and Jack Scott Snoeyink. Approximating polygons and subdivisions with minimum-link paths. International Journal of Computational Geometry & Applications, 03(04):383-415, 1993.
17. John Hershberger and Jack Snoeyink. An O(n log n) implementation of the Douglas-Peucker algorithm for line simplification. In Proceedings 10th Annual ACM Symposium on Computational Geometry, SCG '94, pages 383-384, 1994.
18. Hiroshi Imai and Masao Iri. Polygonal approximations of a curve - formulations and algorithms. In Godfried T. Toussaint, editor, Computational Morphology: A Computational Geometric Approach to the Analysis of Form. North-Holland, Amsterdam, 1988.
19. V. S. Anil Kumar, Sunil Arya, and H. Ramesh. Hardness of set cover with intersection 1. In Automata, Languages and Programming: 27th International Colloquium, ICALP 2000, pages 624-635. Springer, Berlin, Heidelberg, 2000.
20. Nimrod Megiddo and Arie Tamir. On the complexity of locating linear facilities in the plane. Operations Research Letters, 1(5):194-197, 1982.
21. Avraham Melkman and Joseph O'Rourke. On polygonal chain approximation. In Godfried T. Toussaint, editor, Computational Morphology: A Computational Geometric Approach to the Analysis of Form, pages 87-95. North-Holland, Amsterdam, 1988.
22. E. R. Scheinerman. Intersection Classes and Multiple Intersection Parameters of Graphs. PhD thesis, Princeton University, 1984.
23. Marc van Kreveld, Maarten Löffler, and Lionov Wiratma. On optimal polyline simplification using the hausdorff and fréchet distance. URL: https://arxiv.org/abs/1803.03550.
X

Feedback for Dagstuhl Publishing