Gromov-Frechet distance between curves

O. Berezsky1, M. Zarichnyi2
1) Ternopil National Economic University, Ternopil, Ukraine; 2) Department of Mechanics and Mathematics Lviv National University, Lviv, Ukraine
The Gromov-Frechet distance is obtained from the Frechet distance between metric curves similarly as the Gromov-Hausdorff distance is obtained from the Hausdorff distance. We prove that the Gromov-Frechet space is separable and non-complete.
Gromov-Frechet distance; Gromov-Frechet space; Gromov-Hausdorff distance
1. P.K. Agarwal, R.B. Avraham, H. Kaplan, M. Sharir, Computing the discrete FrLechet distance in subquadratic time, SIAM J. Comput., 43 (2014), 2, 429-449.

2. J. Alber, R. Niedermeier, On multidimensional curves with Hilbert property, Theory of Computing Systems, 33 (2000), 4, 295-312.

3. H. Alt, M. Buchin, Can we compute the similarity between surfaces? Discrete Comput. Geom., 43 (2010), 1, 78.99.

4. Alt H., Godau M. Computing the Frechet distance between two polygonal curves, Int. J. of Computational Geometry and Applications, 5 (1995), 75-91.

5. D. Burago, Yu. Burago, S. Ivanov, A course in metric geometry, AMS GSM 33, 2001.

6. D.A. Edwards, The structure of superspace, published in: Studies in Topology, Academic Press, 1975, 121-133.

7. M. Gromov, Groups of polynomial growth and expanding maps, Publications mathematiques I.H.E.S., 53, 1981.

8. M. FrLechet, Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Mathematico di Palermo, 22 (1906), 1-74.

9. G.M. Ewing, Calculus of variations with applications, Dover Publ., 1985.

10. G. Rote, Computing the Frechet distance between piecewise smooth curves, Computational Geometry, 37 (2007), 162-174.

11. M.I. Schlesinger, E.V. Vodolazskiy, V.M. Yakovenko, Frechet similarity of closed polygonal curves, International Journal of Computational Geometry & Applications, 26, (2016), 1, 53-66.

12. C. Kuratowski, Quelques probl`emes concernant les espaces metriques non-separables, Fundamenta Mathematicae, 25 (1935), 534-545.

13. A. Mosig, M. Clausen, Approximately matching polygonal curves with respect to the FrLechet distance, Computational Geometry, 30 (2005), 113-127.

14. T. Eiter, H. Mannila, Computing discrete Frechet distance, Technical Report CDTR 94/64, Christian Doppler Laboratory for Expert Systems, TU Vienna, Austria, 1994.

15. Hee-Kap Ahn, Christian Knauer, Marc Scherfenberg, Lena Schlipf, Antoine Vigneron, Computing the discrete Frechet distance with imprecise input, International Journal of Computational Geometry & Applications, 22 (2012), 1, 27-44.

16. K. Buchin, M. Buchin, C. Wenk, Computing the Frechet distance between simple polygons, Computational Geometry, 41 (2008), 2-20.

17. A.F. Cook IV, A. Driemel, J. Sherette, C.Wenk, Computing the Frechet distance between folded polygons, Computational Geometry, 50 (2015), 1-16.

18. C. Villani, Topics in optimal transportation, Amer. Math. Soc., Providence, Rhode Island, 2000.

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