The Black-Box Complexity of Nearest Neighbor Search

  • Robert Krauthgamer
  • James R. Lee
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3142)

Abstract

We define a natural notion of efficiency for approximate nearest-neighbor (ANN) search in general n-point metric spaces, namely the existence of a randomized algorithm which answers (1+ε)-approximate nearest neighbor queries in polylog(n) time using only polynomial space. We then study which families of metric spaces admit efficient ANN schemes in the black-box model, where only oracle access to the distance function is given, and any query consistent with the triangle inequality may be asked.

For \(\varepsilon < \frac{2}{5}\), we offer a complete answer to this problem. Using the notion of metric dimension defined in [GKL03] (à la [Ass83]), we show that a metric space X admits an efficient (1+ε)-ANN scheme for any \(\varepsilon < \frac{2}{5}\)if and only if\(\dim(X) = O(\log \log n)\). For coarser approximations, clearly the upper bound continues to hold, but there is a threshold at which our lower bound breaks down—this is precisely when points in the “ambient space” may begin to affect the complexity of “hard” subspaces SX. Indeed, we give examples which show that \(\dim(X)\) does not characterize the black-box complexity of ANN above the threshold.

Our scheme for ANN in low-dimensional metric spaces is the first to yield efficient algorithms without relying on any additional assumptions on the input. In previous approaches (e.g., [Cla99,KR02,KL04]), even spaces with \(\dim(X) = O(1)\) sometimes required Ω(n) query times.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ass83]
    Assouad, P.: Plongements lipschitziens dans Rn. Bull. Soc. Math. France 111(4), 429–448 (1983)MATHMathSciNetGoogle Scholar
  2. [Cla99]
    Clarkson, K.L.: Nearest neighbor queries in metric spaces. Discrete Comput. Geom. 22(1), 63–93 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. [GKL03]
    Gupta, A., Krauthgamer, R., Lee, J.R.: Bounded geometries, fractals, and low-distortion embeddings. In: Proceedings of the 44th annual Symposium on the Foundations of Computer Science (2003)Google Scholar
  4. [Gro99]
    Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. Birkhäuser, Boston (1999)MATHGoogle Scholar
  5. [HKMR04]
    Hildrum, K., Kubiatowicz, J., Ma, S., Rao, S.: A note on finding nearest neighbors in growth-restricted metrics. In: Proceedings of the 15th annual ACM-SIAM Symposium on Discrete Algorithms (2004)Google Scholar
  6. [H01]
    Har-Peled, S.: A replacement for Voronoi diagrams of near linear size. In: 42nd IEEE Symposium on Foundations of Computer Science, Las Vegas, NV, pp. 94–103. IEEE Computer Soc., Los Alamitos (2001)Google Scholar
  7. [IM98]
    Indyk, P., Motwani, R.: Approximate nearest neighbors: towards removing the curse of dimensionality. In: 30th Annual ACM Symposium on Theory of Computing, May 1998, pp. 604–613 (1998)Google Scholar
  8. [KKL03]
    Kakade, S., Kearns, M., Langford, J.: Exploration in metric state spaces. In: Proc. of the 20th International Conference on Machine Learning (2003)Google Scholar
  9. [KL04]
    Krauthgamer, R., Lee, J.R.: Navigating nets: Simple algorithms for proximity search. In: Proceedings of the 15th annual ACM-SIAM Symposium on Discrete Algorithms (2004)Google Scholar
  10. [KOR98]
    Kushilevitz, E., Ostrovsky, R., Rabani, Y.: Efficient search for approximate nearest neighbor in high dimensional spaces. In: 30th Annual ACM Symposium on the Theory of Computing, pp. 614–623 (1998)Google Scholar
  11. [KR02]
    Karger, D., Ruhl, M.: Finding nearest neighbors in growth-restricted metrics. In: 34th Annual ACM Symposium on the Theory of Computing, pp. 63–66 (2002)Google Scholar
  12. [Tal04]
    Talwar, K.: Bypassing the embedding: Approximation schemes and distance labeling schemes for growth restricted metrics. To appear in the procedings of the 36th annual Symposium on the Theory of Computing (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Robert Krauthgamer
    • 1
  • James R. Lee
    • 2
  1. 1.IBM Almaden Research CenterSan JoseUSA
  2. 2.Computer Science DivisionU.C. BerkeleyBerkeleyUSA

Personalised recommendations