On approximate nearest neighbors in non-Euclidean spaces

The nearest neighbor search (NNS) problem is the following: Given a set of n points P = {p(1),..., p(n)} in some metric space X preprocess P so as to efficiently answer queries which require finding a point in P closest to a queries point q is an element of 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 (4 log(1+p) log 4d + 3)-NNS algorithm in l(infinity)(d) with O(dn(1+p) log n) storage and O(d log n) query time. In particular this fields the first algorithm for O(1)-NNS for l(infinity) with subexponential storage. The preprocessing time is linear in the size of the data structure. The algorithm call be also used (after simple modifications) to output the exact nearest neighbor in time bounded by O(d log n) plus the number of (4 log(1+p) log 4d + 3)-nearest neighbors of the query point. Building on this result, we also obtain mt approximation algorithm for a general class of product metrics. Finally, we show that for any c < 3 the c-NNS problem in l(infinity) is provably hard for a version of the indexing model introduced by Hellerstein et. al. [HKP97](our upper bound call be adapted to work in this model).