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An Algorithm for Reducing Approximate Nearest Neighbor to Approximate Near Neighbor with \(O(\log {n})\) Query Time

  • Hengzhao Ma
  • Jianzhong Li
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

Abstract

This paper proposes a new algorithm for reducing Approximate Nearest Neighbor problem to Approximate Near Neighbor problem. The advantage of this algorithm is that it achieves \(O(\log {n})\) query time. As a reduction problem, the query time complexity is the times of invoking the algorithm for Approximate Near Neighbor problem. All former algorithms for the same reduction need polylog(n) query time. A box split method proposed by Vaidya is used in our paper to achieve the \(O(\log {n})\) query time complexity.

Keywords

Computation geometry Approximate nearest neighbor Reduction 

References

  1. 1.
    Andoni, A., Indyk, P.: Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. In: 47th Annual IEEE Symposium on Foundations of Computer Science, vol. 51, pp. 459–468 (2006)Google Scholar
  2. 2.
    Andoni, A., Indyk, P.: Nearest neighbors in high-dimensional spaces. In: Handbook of Discrete and Computational Geometry, 3rd edn, chap. 43, pp. 1135–1155. CRC Press Inc., Boca Raton (2017)Google Scholar
  3. 3.
    Andoni, A., Razenshteyn, I.: Optimal data-dependent hashing for approximate near neighbors. In: Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, pp. 793–801 (2015)Google Scholar
  4. 4.
    Arya, S., Mount, D.M.: Approximate nearest neighbor queries in fixed dimensions. In: Proceedings of the Fourth Annual Symposium on Discrete Algorithms, pp. 271–280 (1993)Google Scholar
  5. 5.
    Arya, S., Mount, D.M., Netanyahu, N.S., Silverman, R., Wu, A.Y.: An optimal algorithm for approximate nearest neighbor searching fixed dimensions. J. ACM 45(6), 891–923 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bern, M.W.: Approximate closest-point queries in high dimensions. Inf. Process. Lett. 45(2), 95–99 (1993)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Callahan, P.B., Kosaraju, S.R.: A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM 42(1), 67–90 (1995)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chan, T.M.: Approximate nearest neighbor queries revisited. In: Proceedings of the Thirteenth Annual Symposium on Computational Geometry, vol. 20, pp. 352–358 (1997)Google Scholar
  9. 9.
    Datar, M., Immorlica, N., Indyk, P., Mirrokni, V.S.: Locality-sensitive hashing scheme based on p-table distributions. In: Proceedings of the Twentieth annual Symposium on Computational Geometry, pp. 253–262 (2004)Google Scholar
  10. 10.
    Feder, T., Greene, D.: Optimal algorithms for approximate clustering. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 434–444 (1988)Google Scholar
  11. 11.
    Goel, A., Indyk, P., Varadarajan, K.: Reductions among high dimensional proximity problems. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 769–778 (2001)Google Scholar
  12. 12.
    Har-Peled, S.: A replacement for voronoi diagrams of near linear size. In: 42nd Annual IEEE Symposium on Foundations of Computer Science, pp. 94–103 (2001)Google Scholar
  13. 13.
    Har-Peled, S., Indyk, P., Motwani, R.: Approximate nearest neighbor: towards removing the curse of dimensionality. Theory Comput. 8(1), 321–350 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Indyk, P., Motwani, R.: Approximate nearest neighbors: towards removing the curse of dimensionality. In: Proceedings of the Thirteenth Annual ACM Symposium on Theory of Computing, pp. 604–613 (1998)Google Scholar
  15. 15.
    Kleinberg, J.M.: Two algorithms for nearest-neighbor search in high dimensions. In: Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, pp. 599–608 (1997)Google Scholar
  16. 16.
    Krauthgamer, R., Lee, J.R.: Navigating nets: simple algorithms for proximity search. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 798–807 (2004)Google Scholar
  17. 17.
    Kushilevitz, E., Ostrovsky, R., Rabani, Y.: Efficient search for approximate nearest neighbor in high dimensional spaces. SIAM J. Comput. 30(2), 457–474 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ma, H., Li, J.: An Algorithm for Reducing Approximate Nearest Neighbor to Approximate Near Neighbor with O(logn) Query Time (2018). http://arxiv.org/abs/1809.09776
  19. 19.
    Panigrahy, R.: Entropy based nearest neighbor search in high dimensions. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1186–1195 (2006)Google Scholar
  20. 20.
    Vaidya, P.M.: An optimal algorithm for the all-nearest-neighbors problem. In: 27th Annual Symposium on Foundations of Computer Science, pp. 117–122 (1986)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Harbin Institute of TechnologyHarbinChina

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