Discrete & Computational Geometry

, Volume 48, Issue 1, pp 94–127 | Cite as

Approximating the Fréchet Distance for Realistic Curves in Near Linear Time

Article

Abstract

We present a simple and practical (1+ε)-approximation algorithm for the Fréchet distance between two polygonal curves in ℝd. To analyze this algorithm we introduce a new realistic family of curves, c-packed curves, that is closed under simplification. We believe the notion of c-packed curves to be of independent interest. We show that our algorithm has near linear running time for c-packed polygonal curves, and similar results for other input models, such as low-density polygonal curves.

Keywords

Frechet distance Approximation algorithms Realistic input models 

References

  1. 1.
    Agarwal, P.K., Har-Peled, S., Mustafa, N., Wang, Y.: Near-linear time approximation algorithms for curve simplification in two and three dimensions. Algorithmica 42, 203–219 (2005) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Alt, H.: The computational geometry of comparing shapes. In: Efficient Algorithms: Essays Dedicated to Kurt Mehlhorn on the Occasion of His 60th Birthday, pp. 235–248. Springer, Berlin (2009) Google Scholar
  3. 3.
    Alt, H., Godau, M.: Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geom. Appl. 5, 75–91 (1995) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Alt, H., Knauer, C., Wenk, C.: Comparison of distance measures for planar curves. Algorithmica 38(1), 45–58 (2004) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Aronov, B., Har-Peled, S.: On approximating the depth and related problems. SIAM J. Comput. 38(3), 899–921 (2008) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Aronov, B., Har-Peled, S., Knauer, C., Wang, Y., Wenk, C.: Fréchet distance for curves, revisited. In: Proc. 14th Annu. European Symp. Algorithms, pp. 52–63 (2006) Google Scholar
  7. 7.
    Bose, P., Cheong, O., Dujmović, V.: A note on the perimeter of fat objects. Comput. Geom. Theory Appl. 44(1), 1–8 (2011) MATHCrossRefGoogle Scholar
  8. 8.
    Brakatsoulas, S., Pfoser, D., Salas, R., Wenk, C.: On map-matching vehicle tracking data. In: Proc. 31st VLDB Conference, pp. 853–864 (2005) Google Scholar
  9. 9.
    Buchin, K., Buchin, M., Knauer, C., Rote, G., Wenk, C.: How difficult is it to walk the dog. In: Proc. 23rd Euro. Workshop on Comput. Geom., pp. 170–173 (2007) Google Scholar
  10. 10.
    Buchin, K., Buchin, M., Gudmundsson, J.: Detecting single file movement. In: Proc. 16th ACM SIGSPATIAL Int. Conf. Adv. GIS, pp. 288–297 (2008) Google Scholar
  11. 11.
    Buchin, K., Buchin, M., Gudmundsson, J., Maarten, L., Luo, J.: Detecting commuting patterns by clustering subtrajectories. In: Proc. 19th Annu. Internat. Symp. Algorithms Comput., pp. 644–655 (2008) Google Scholar
  12. 12.
    Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. Assoc. Comput. Mach. 42, 67–90 (1995) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    de Berg, M.: Linear size binary space partitions for uncluttered scenes. Algorithmica 28, 353–366 (2000) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    de Berg, M.: Improved bounds on the union complexity of fat objects. Discrete Comput. Geom. 40(1), 127–140 (2008) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    de Berg, M., Katz, M.J., van der Stappen, A.F., Vleugels, J.: Realistic input models for geometric algorithms. Algorithmica 34, 81–97 (2002) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Driemel, A., Har-Peled, S., Wenk, C.: Approximating the Fréchet distance for realistic curves in near linear time. In: Proc. 26th Annu. ACM Symp. Comput. Geom., pp. 365–374 (2010). arXiv:1003.0460 Google Scholar
  17. 17.
    Efrat, A.: The complexity of the union of (α,β)-covered objects. SIAM J. Comput. 34(4), 775–787 (2005) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Erickson, J.: On the relative complexities of some geometric problems. In: Proc. 7th Canad. Conf. Comput. Geom., pp. 85–90 (1995) Google Scholar
  19. 19.
    Keogh, E.J., Pazzani, M.J.: Scaling up dynamic time warping to massive dataset. In: Proc. of the Third Euro. Conf. Princip. Data Mining and Know. Disc., pp. 1–11 (1999) CrossRefGoogle Scholar
  20. 20.
    Kim, M.S., Kim, S.W., Shin, M.: Optimization of subsequence matching under time warping in time-series databases. In: Proc. ACM Symp. Appl. Comput., pp. 581–586 (2005) CrossRefGoogle Scholar
  21. 21.
    Kwong, S., He, Q.H., Man, K.F., Tang, K.S., Chau, C.W.: Parallel genetic-based hybrid pattern matching algorithm for isolated word recognition. Int. J. Pattern Recognit. Artif. Intell. 12(5), 573–594 (1998) CrossRefGoogle Scholar
  22. 22.
    Munich, M.E., Perona, P.: Continuous dynamic time warping for translation-invariant curve alignment with applications to signature verification. In: Proc. 7th Int. Conf. Comp. Vision, pp. 108–115 (1999) CrossRefGoogle Scholar
  23. 23.
    Sriraghavendra, E., Karthik, K., Bhattacharyya, C.: Fréchet distance based approach for searching online handwritten documents. In: Proc. 9th Int. Conf. Doc. Anal. Recogn., pp. 461–465 (2007) Google Scholar
  24. 24.
    Wenk, C., Salas, R., Pfoser, D.: Addressing the need for map-matching speed: localizing global curve-matching algorithms. In: Proc. 18th Int. Conf. Sci. Statist. Database Managm., pp. 879–888 (2006) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands
  2. 2.Department of Computer ScienceUniversity of IllinoisUrbanaUSA
  3. 3.Department of Computer ScienceUniversity of Texas at San AntonioSan AntonioUSA

Personalised recommendations