International Conference on Similarity Search and Applications

Similarity Search and Applications pp 3-14 | Cite as

Approximate Furthest Neighbor in High Dimensions

  • Rasmus Pagh
  • Francesco Silvestri
  • Johan Sivertsen
  • Matthew Skala
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9371)

Abstract

Much recent work has been devoted to approximate nearest neighbor queries. Motivated by applications in recommender systems, we consider approximate furthest neighbor (AFN) queries. We present a simple, fast, and highly practical data structure for answering AFN queries in high-dimensional Euclidean space. We build on the technique of Indyk (SODA 2003), storing random projections to provide sublinear query time for AFN. However, we introduce a different query algorithm, improving on Indyk’s approximation factor and reducing the running time by a logarithmic factor. We also present a variation based on a query-independent ordering of the database points; while this does not have the provable approximation factor of the query-dependent data structure, it offers significant improvement in time and space complexity. We give a theoretical analysis, and experimental results.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abbar, S., Amer-Yahia, S., Indyk, P., Mahabadi, S.: Real-time recommendation of diverse related articles. In: Proc. 22nd International Conference on World Wide Web (WWW), pp. 1–12 (2013)Google Scholar
  2. 2.
    Bădoiu, M., Clarkson, K.L.: Optimal core-sets for balls. Computational Geometry 40(1), 14–22 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer-Verlag TELOS (2008)Google Scholar
  4. 4.
    Bespamyatnikh, S.N.: Dynamic algorithms for approximate neighbor searching. In: Proceedings of the 8th Canadian Conference on Computational Geometry (CCCG 1996), pp. 252–257. Carleton University, August 12–15, 1996Google Scholar
  5. 5.
    Chávez, E., Navarro, G.: Measuring the dimensionality of general metric spaces. Tech. Rep. TR/DCC-00-1, Department of Computer Science, University of Chile (2000)Google Scholar
  6. 6.
    Clarkson, K.L.: Las Vegas algorithms for linear and integer programming when the dimension is small. Journal of the ACM (JACM) 42(2), 488–499 (1995)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Datar, M., Immorlica, N., Indyk, P., Mirrokni, V.S.: Locality-sensitive hashing scheme based on p-stable distributions. In: Proc. 20 Annual Symposium on Computational Geometry (SoCG), pp. 253–262 (2004)Google Scholar
  8. 8.
    Figueroa, K., Navarro, G., Chávez, E.: Metric spaces library (2007) (online). http://www.sisap.org/Metric_Space_Library.html
  9. 9.
    Goel, A., Indyk, P., Varadarajan, K.: Reductions among high dimensional proximity problems. In: Proc. 12th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 769–778 (2001)Google Scholar
  10. 10.
    Indyk, P.: Better algorithms for high-dimensional proximity problems via asymmetric embeddings. In: Proc. 14th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 539–545 (2003)Google Scholar
  11. 11.
    Indyk, P., Mahabadi, S., Mahdian, M., Mirrokni, V.S.: Composable core-sets for diversity and coverage maximization. In: Proc. 33rd ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS), pp. 100–108. ACM (2014)Google Scholar
  12. 12.
    Karger, D., Motwani, R., Sudan, M.: Approximate graph coloring by semidefinite programming. Journal of the ACM (JACM) 45(2), 246–265 (1998)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kumar, P., Mitchell, J.S., Yildirim, E.A.: Approximate minimum enclosing balls in high dimensions using core-sets. Journal of Experimental Algorithmics 8, 1–1 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Matoušek, J.: On variants of the Johnson-Lindenstrauss lemma. Random Structures and Algorithms 33(2), 142–156 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Matoušek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Algorithmica 16(4–5), 498–516 (1996)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Said, A., Fields, B., Jain, B.J., Albayrak, S.: User-centric evaluation of a k-furthest neighbor collaborative filtering recommender algorithm. In: Proc. Conference on Computer Supported Cooperative Work (CSCW), pp. 1399–1408 (2013)Google Scholar
  17. 17.
    Said, A., Kille, B., Jain, B.J., Albayrak, S.: Increasing diversity through furthest neighbor-based recommendation. In: Proceedings of the WSDM Workshop on Diversity in Document Retrieval (DDR 2012) (2012)Google Scholar
  18. 18.
    Skala, M.A.: Aspects of Metric Spaces in Computation. Ph.D. thesis, University of Waterloo (2008)Google Scholar
  19. 19.
    Williams, R.: A new algorithm for optimal constraint satisfaction and its implications. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1227–1237. Springer, Heidelberg (2004) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Rasmus Pagh
    • 1
  • Francesco Silvestri
    • 1
  • Johan Sivertsen
    • 1
  • Matthew Skala
    • 1
  1. 1.IT University of CopenhagenCopenhagenDenmark

Personalised recommendations