The Black-Box Complexity of Nearest Neighbor Search

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The Black-Box Complexity of Nearest Neighbor Search
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 ε < 2 5 , we offer a complete answer to this problem. Using the notion of metric dimension defined in [GKL03] (`a la [Ass83]), we show that a metric space X admits an efficient (1+ε)-ANN scheme for any ε < 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 ...
Robert Krauthgamer, James R. Lee
Added 01 Jul 2010
Updated 01 Jul 2010
Type Conference
Year 2004
Authors Robert Krauthgamer, James R. Lee
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