The black-box complexity of nearest-neighbor search

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 + ɛ ) -ANN 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 ɛ < 2 5 , we offer a complete answer to this problem. Using the notion of metric dimension defined in [A. Gupta, et al., Bounded geometries, fractals, and low-distortion embeddings, in: 44th Annu. IEEE Symp. on Foundations of Computer Science, 2003, pp. 534–543] (à la [P. Assouad, Plongements lipschitziens dans R n , Bull. Soc. Math. France 111 (4) (1983) 429–448]), we show that a metric space X admits an efficient ( 1 + ɛ ) -ANN scheme for any ɛ < 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 S ⊆ X . 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., [K.L. Clarkson, Nearest neighbor queries in metric spaces, Discrete Comput. Geom. 22(1) (1999) 63–93; D. Karger, M. Ruhl, Finding nearest neighbors in growth-restricted metrics, in: 34th Annu. ACM Symp. on the Theory of Computing, 2002, pp. 63–66; R. Krauthgamer, J.R. Lee, Navigating nets: simple algorithms for proximity search, in: 15th Annu. ACM-SIAM Symp. on Discrete Algorithms, 2004, pp. 791–801; K. Hildrum, et al., A note on finding nearest neighbors in growth-restricted metrics, in: Proc. of the 15th Annu. ACM-SIAM Symp. on Discrete Algorithms, 2004, pp. 560–561]), even spaces with dim ( X ) = O ( 1 ) sometimes required Ω ( n ) query times.