Computing the Fréchet Gap Distance
Measuring the similarity of two polygonal curves is a fundamental computational task. Among alternatives, the Frechet distance is one of the most well studied similarity measures. Informally, the Fréchet distance is described as the minimum leash length required for a man on one of the curves to walk a dog on the other curve continuously from the starting to the ending points. In this paper we study a variant called the Fréchet gap distance. In the man and dog analogy, the Fréchet gap distance minimizes the difference of the longest and smallest leash lengths used over the entire walk. This measure in some ways better captures our intuitive notions of curve similarity, for example giving distance zero to translated copies of the same curve.
The Fréchet gap distance was originally introduced by Filtser and Katz (2015) in the context of the discrete Fréchet distance. Here we study the continuous version, which presents a number of additional challenges not present in discrete case. In particular, the continuous nature makes bounding and searching over the critical events a rather difficult task.
For this problem we give an O(n^5 log(n)) time exact algorithm and a more efficient O(n^2 log(n) + (n^2/epsilon) log(1/epsilon)) time (1+epsilon)-approximation algorithm, where n is the total number of vertices of the input curves. Note that for (small enough) constant epsilon and ignoring logarithmic factors, our approximation has quadratic running time, matching the lower bound, assuming SETH (Bringmann 2014), for approximating the standard Fréchet distance for general curves.
Frechet Distance
Approximation
Polygonal Curves
42:1-42:16
Regular Paper
Chenglin
Fan
Chenglin Fan
Benjamin
Raichel
Benjamin Raichel
10.4230/LIPIcs.SoCG.2017.42
P. Agarwal, R. Avraham, H. Kaplan, and M. Sharir. Computing the discrete Fréchet distance in subquadratic time. SIAM Journal on Computing, 43(2):429-449, 2014.
H. Alt and M. Buchin. Can we compute the similarity between surfaces? Discrete &Computational Geometry, 43(1):78-99, 2010.
H. Alt, A. Efrat, G. Rote, and C. Wenk. Matching planar maps. In Proc. of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 589-598, 2003.
H. Alt and M. Godau. Computing the Fréchet distance between two polygonal curves. Int. J. Comput. Geometry Appl., 5:75-91, 1995.
H. Alt, C. Knauer, and C. Wenk. Matching polygonal curves with respect to the Fréchet distance. In Annual Symp. on Theo. Aspects of Comp. Sci. (STACS), pages 63-74, 2001.
R. Avraham, O. Filtser, H. Kaplan, M. Katz, and M. Sharir. The discrete and semicontinuous Fréchet distance with shortcuts via approximate distance counting and selection. ACM Trans. Algorithms, 11(4):29, 2015.
R. Avraham, H. Kaplan, and M. Sharir. A faster algorithm for the discrete Fréchet distance under translation. CoRR, abs/1501.03724, 2015.
J. Bentley and T. Ottmann. Algorithms for reporting and counting geometric intersections. IEEE Trans. Computers, 28(9):643-647, 1979.
S. Brakatsoulas, D. Pfoser, R. Salas, and C. Wenk. On map-matching vehicle tracking data. In Proc. 31st VLDB Conference, pages 853-864, 2005.
K. Bringmann. Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless seth fails. In Symp. on Found. of Comp. Sci. (FOCS), pages 661-670. IEEE, 2014.
K. Buchin, M. Buchin, J. Gudmundsson, M. Löffler, and J. Luo. Detecting commuting patterns by clustering subtrajectories. Int. J. Comput. Geom. Appl., 21(3):253-282, 2011.
K. Buchin, M. Buchin, W. Meulemans, and W. Mulzer. Four soviets walk the dog - with an application to alt’s conjecture. In Proc. of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1399-1413, 2014.
K. Buchin, M. Buchin, and C. Wenk. Computing the Fréchet distance between simple polygons in polynomial time. In 22nd Annual Symp. Comput. Geom., pages 80-87, 2006.
M. Buchin, A. Driemel, and B. Speckmann. Computing the Fréchet distance with shortcuts is np-hard. In 30th Annual Symp. Comput. Geom. (SoCG), page 367, 2014.
A. Driemel and S. Har-Peled. Jaywalking your dog: computing the Fréchet distance with shortcuts. SIAM Journal on Computing, 42(5):1830-1866, 2013.
A. Driemel, S. Har-Peled, and C. Wenk. Approximating the Fréchet distance for realistic curves in near linear time. Discrete &Computational Geometry, 48(1):94-127, 2012.
T. Eiter and H. Mannila. Computing discrete Fréchet distance, 1994.
C. Fan and B. Raichel. Computing the Fréchet gap distance. URL: http://www.utdallas.edu/~bar150630/gap.pdf.
http://www.utdallas.edu/~bar150630/gap.pdf
O. Filtser and M. Katz. The discrete Fréchet distance gap. arXiv:1506.04861, 2015.
S. Har-Peled and B. Raichel. The Fréchet distance revisited and extended. ACM Transactions on Algorithms (TALG), 10(1):3, 2014.
M. Kim, S. Kim, and M. Shin. Optimization of subsequence matching under time warping in time-series databases. In Proc. ACM Symp. on Applied Computing, pages 581-586, 2005.
G. Rote. Computing the Fréchet distance between piecewise smooth curves. Computational Geometry, 37(3):162-174, 2007.
J. Serrà, E. Gómez, P. Herrera, and X. Serra. Chroma binary similarity and local alignment applied to cover song identifica. Audio, Speech & Lang. Proc., 16(6):1138-1151, 2008.
C. Wenk, R. Salas, and D. Pfoser. Addressing the need for map-matching speed: Localizing global curve-matching algorithms. In Sci. Statis. Database Manag., pages 879-888, 2006.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode