On Approximate Nearest Neighbors in Non-Euclidean Spaces

The nearest neighbor search (NNS) problem is the following: Given a set of n points P = fp 1 ; : : : ; p n g in some metric space X , preprocess P so as to eeciently answer queries which require nding a point in P closest to a query point q 2 X. The approximate nearest neighbor search (c-NNS) is a relaxation of NNS which allows to return any point within c times the distance to the nearest neighbor (called c-nearest neighbor). This problem is of major and growing importance to a variety of applications. In this paper, we give an algorithm for (4dlog 1+ log 4de + 3)-NNS algorithm in l d 1 with O(dn 1+ log n) storage and O(d log n) query time. In particular, this yields the rst algorithm for O(1)-NNS for l 1 with subexponential storage. The preprocessing time is close to linear in the size of the data structure. The algorithm can be also used (after simple modiications) to output the exact nearest neighbor in time bounded by O(d logn) plus the number of (4dlog 1+ log 4de + 3)-nearest neighbors of the query point. Building on this result, we also obtain an approximation algorithm for a general class of product metrics. Finally, we show that for any c < 3 the c-NNS problem in l 1 is prov-ably hard for a version of the indexing model introduced by Hellerstein et. al. HKP97] (our upper bound can be adapted to work in this model).