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+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 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., [Cla99,KR02,KL04]), even spaces with $\dim(X) = O(1)$ sometimes required Ω(n) query times.