Algorithms for the Discrete Fréchet Distance Under Translation

Authors Omrit Filtser, Matthew J. Katz

Thumbnail PDF


  • Filesize: 0.71 MB
  • 14 pages

Document Identifiers

Author Details

Omrit Filtser
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Matthew J. Katz
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

Cite AsGet BibTex

Omrit Filtser and Matthew J. Katz. Algorithms for the Discrete Fréchet Distance Under Translation. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


The (discrete) Fréchet distance (DFD) is a popular similarity measure for curves. Often the input curves are not aligned, so one of them must undergo some transformation for the distance computation to be meaningful. Ben Avraham et al. [Rinat Ben Avraham et al., 2015] presented an O(m^3n^2(1+log(n/m))log(m+n))-time algorithm for DFD between two sequences of points of sizes m and n in the plane under translation. In this paper we consider two variants of DFD, both under translation. For DFD with shortcuts in the plane, we present an O(m^2n^2 log^2(m+n))-time algorithm, by presenting a dynamic data structure for reachability queries in the underlying directed graph. In 1D, we show how to avoid the use of parametric search and remove a logarithmic factor from the running time of (the 1D versions of) these algorithms and of an algorithm for the weak discrete Fréchet distance; the resulting running times are thus O(m^2n(1+log(n/m))), for the discrete Fréchet distance, and O(mn log(m+n)), for its two variants. Our 1D algorithms follow a general scheme introduced by Martello et al. [Martello et al., 1984] for the Balanced Optimization Problem (BOP), which is especially useful when an efficient dynamic version of the feasibility decider is available. We present an alternative scheme for BOP, whose advantage is that it yields efficient algorithms quite easily, without having to devise a specially tailored dynamic version of the feasibility decider. We demonstrate our scheme on the most uniform path problem (significantly improving the known bound), and observe that the weak DFD under translation in 1D is a special case of it.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • curve similarity
  • discrete Fréchet distance
  • translation
  • algorithms
  • BOP


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


  1. Pankaj K. Agarwal, Rinat Ben Avraham, Haim Kaplan, and Micha Sharir. Computing the discrete fréchet distance in subquadratic time. SIAM J. Comput., 43(2):429-449, 2014. URL:
  2. Helmut Alt and Michael Godau. Computing the fréchet distance between two polygonal curves. Int. J. Comput. Geometry Appl., 5:75-91, 1995. URL:
  3. Helmut Alt, Christian Knauer, and Carola Wenk. Matching polygonal curves with respect to the fréchet distance. In Afonso Ferreira and Horst Reichel, editors, STACS 2001, 18th Annual Symposium on Theoretical Aspects of Computer Science, Dresden, Germany, February 15-17, 2001, Proceedings, volume 2010 of Lecture Notes in Computer Science, pages 63-74. Springer, 2001. URL:
  4. Rinat Ben Avraham, Omrit Filtser, Haim Kaplan, Matthew J. Katz, and Micha Sharir. The discrete fréchet distance with shortcuts via approximate distance counting and selection. In Siu-Wing Cheng and Olivier Devillers, editors, 30th Annual Symposium on Computational Geometry, SOCG'14, Kyoto, Japan, June 08 - 11, 2014, page 377. ACM, 2014. URL:
  5. Rinat Ben Avraham, Haim Kaplan, and Micha Sharir. A faster algorithm for the discrete fréchet distance under translation. CoRR, abs/1501.03724, 2015. URL:
  6. Sotiris Brakatsoulas, Dieter Pfoser, Randall Salas, and Carola Wenk. On map-matching vehicle tracking data. In Klemens Böhm, Christian S. Jensen, Laura M. Haas, Martin L. Kersten, Per-Åke Larson, and Beng Chin Ooi, editors, Proceedings of the 31st International Conference on Very Large Data Bases, Trondheim, Norway, August 30 - September 2, 2005, pages 853-864. ACM, 2005. URL:
  7. Karl Bringmann. Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 661-670. IEEE Computer Society, 2014. URL:
  8. Kevin Buchin, Maike Buchin, Wouter Meulemans, and Wolfgang Mulzer. Four soviets walk the dog - with an application to alt’s conjecture. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1399-1413. SIAM, 2014. URL:
  9. Kevin Buchin, Maike Buchin, and Yusu Wang. Exact algorithms for partial curve matching via the fréchet distance. In Claire Mathieu, editor, Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009, pages 645-654. SIAM, 2009. URL:
  10. Maike Buchin. On the computability of the Fréchet distance between triangulated surfaces. PhD thesis, FU Berlin, 2007. Google Scholar
  11. Maike Buchin, Anne Driemel, and Bettina Speckmann. Computing the fréchet distance with shortcuts is np-hard. In Siu-Wing Cheng and Olivier Devillers, editors, 30th Annual Symposium on Computational Geometry, SOCG'14, Kyoto, Japan, June 08 - 11, 2014, page 367. ACM, 2014. URL:
  12. Anne Driemel and Sariel Har-Peled. Jaywalking your dog: Computing the fréchet distance with shortcuts. SIAM J. Comput., 42(5):1830-1866, 2013. URL:
  13. Alon Efrat, Quanfu Fan, and Suresh Venkatasubramanian. Curve matching, time warping, and light fields: New algorithms for computing similarity between curves. Journal of Mathematical Imaging and Vision, 27(3):203-216, 2007. URL:
  14. Thomas Eiter and Heikki Mannila. Computing discrete Fréchet distance. Technical Report CD-TR 94/64, Information Systems Dept., Technical University of Vienna, 1994. Google Scholar
  15. David Eppstein, Giuseppe F. Italiano, Roberto Tamassia, Robert Endre Tarjan, Jeffery Westbrook, and Moti Yung. Maintenance of a minimum spanning forest in a dynamic plane graph. J. Algorithms, 13(1):33-54, 1992. URL:
  16. Omrit Filtser and Matthew J. Katz. The discrete fréchet gap. CoRR, abs/1506.04861, 2015. URL:
  17. Zvi Galil and Baruch Schieber. On finding most uniform spanning trees. Discrete Applied Mathematics, 20(2):173-175, 1988. URL:
  18. Pierre Hansen, Giovanni Storchi, and Tsevi Vovor. Paths with minimum range and ratio of arc lengths. Discrete Applied Mathematics, 78(1-3):89-102, 1997. URL:
  19. Jacob Holm, Kristian de Lichtenberg, and Mikkel Thorup. Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM, 48(4):723-760, 2001. URL:
  20. Minghui Jiang, Ying Xu, and Binhai Zhu. Protein structure-structure alignment with discrete fréchet distance. J. Bioinformatics and Computational Biology, 6(1):51-64, 2008. URL:
  21. Silvano Martello, WR Pulleyblank, Paolo Toth, and Dominique De Werra. Balanced optimization problems. Operations Research Letters, 3(5):275-278, 1984. Google Scholar
  22. Axel Mosig and Michael Clausen. Approximately matching polygonal curves with respect to the fre'chet distance. Comput. Geom., 30(2):113-127, 2005. URL:
  23. Daniel Dominic Sleator and Robert Endre Tarjan. A data structure for dynamic trees. J. Comput. Syst. Sci., 26(3):362-391, 1983. URL:
  24. Carola Wenk. Shape matching in higher dimensions. PhD thesis, Free University of Berlin, Dahlem, Germany, 2003. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail