Space-Efficient Approximation Scheme for Circular Earth Mover Distance

  • Joshua Brody
  • Hongyu Liang
  • Xiaoming Sun
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)


The Earth Mover Distance (EMD) between point sets A and B is the minimum cost of a bipartite matching between A and B. EMD is an important measure for estimating similarities between objects with quantifiable features and has important applications in several areas including computer vision. The streaming complexity of approximating EMD between point sets in a two-dimensional discretized grid is an important open problem proposed in [8,9].

We study the problem of approximating EMD in the streaming model, when points lie on a discretized circle. Computing the EMD in this setting has applications to computer vision [13] and can be seen as a special case of computing EMD on a discretized grid. We achieve a (1 ±ε) approximation for EMD in \(\tilde O(\varepsilon^{-3})\) space, for every 0 < ε < 1. To our knowledge, this is the first streaming algorithm for a natural and widely applied EMD model that matches the space bound asked in [9].


Approximation Scheme Bipartite Match Important Open Problem Constant Factor Approximation Algorithm Stream Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. Journal of Computer and System Sciences 58(1), 137–147 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Andoni, A., Do Ba, K., Indyk, P., Woodruff, D.P.: Efficient sketches for earth-mover distance, with applications. In: Proceedings of the 50th Annual Symposium on Foundations of Computer Science, FOCS (2009)Google Scholar
  3. 3.
    Cabrelli, C.A., Molter, U.M.: A linear time algorithm for a matching problem on the circle. Information Processing Letters 66(3), 161–164 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Grauman, K., Darrell, T.: Fast contour matching using approximate Earth Movers distance. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, CVPR (2004)Google Scholar
  5. 5.
    Grauman, K., Darrell, T.: Efficient image matching with distributions of local invariant features. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, CVPR (2005)Google Scholar
  6. 6.
    Kane, D.M., Nelson, J., Woodruff, D.P.: On the exact space complexity of sketching and streaming small norms. In: Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms, SODA (2010)Google Scholar
  7. 7.
    Lowe, D.G.: Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision 60(2), 91–110 (2004)CrossRefGoogle Scholar
  8. 8.
    McGregor, A.: Open problems in data streams and related topics. In: IITK Workshop on Algorithms For Data Streams (2006),
  9. 9.
    McGregor, A.: Open problems in data streams, property testing, and related topics. In: Bernitoro Workshop on Sublinear Algorithms (2011)Google Scholar
  10. 10.
    Munro, J.I., Paterson, M.: Selection and sorting with limited storage. Theoretical Computer Science 12(3), 315–323 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Muthukrishnan, S.: Data streams: Algorithms and applications. Foundations and Trends in Theoretical Computer Science 1(2), 117–236 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Naor, A., Schechtman, G.: Planar earthmover is not in L 1. SIAM Journal on Computing 37(3), 804–826 (2007); Preliminary version in FOCS 2006MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Rabin, J., Delon, J., Gousseau, Y.: Circular earth mover’s distance for the comparison of local features. In: Proceedings of the IEEE International Conference on Pattern Recognition (ICPR). IEEE Computer Society (2008)Google Scholar
  14. 14.
    Rabin, J., Delon, J., Gousseau, Y.: A statistical approach to the matching of local features. SIAM Journal on Imaging Sciences 2(3), 931–958 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Rabin, J., Delon, J., Gousseau, Y.: Transportation distances on the circle. Journal of Mathematical Imaging and Vision 41(1-2), 147–167 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Rubner, Y., Tomassi, C., Guibas, L.J.: A metric for distributions with applications to image databases. In: Proceedings of the 6th International Conference on Computer Vision, ICCV (1998)Google Scholar
  17. 17.
    Rubner, Y., Tomassi, C., Guibas, L.J.: The earth mover’s distance as a metric for image retrieval. International Journal of Computer Vision 40(2), 99–121 (2000)zbMATHCrossRefGoogle Scholar
  18. 18.
    Venkatesh Babu, R., Pérez, P., Bouthemy, P.: Robust tracking with motion estimation and local kernel-based color modeling. Image and Vision Computing 25(8), 1205–1216 (2007)CrossRefGoogle Scholar
  19. 19.
    Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics, vol. 58. American Math. Soc. (2003)Google Scholar
  20. 20.
    Werman, M., Peleg, S., Melter, R., Kong, T.Y.: Bipartite graph matching for points on a line or a circle. Journal of Algorithms 7(2), 277–284 (1986)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Joshua Brody
    • 1
  • Hongyu Liang
    • 2
  • Xiaoming Sun
    • 3
  1. 1.Aarhus UniversityAarhusDenmark
  2. 2.Institute for Interdisciplinary Information SciencesTsinghua UniversityChina
  3. 3.Institute of Computing TechnologyChinese Academy of SciencesChina

Personalised recommendations