On approximate nearest neighbors in non-Euclidean spaces

The nearest neighbor search (NNS) problem is the following: Given a set of n points P={p₁,...,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 query point q∈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 (4log_1+ρlog4d+3)-NNS algorithm in l_∞^d with O(dn^1+ρlogn) storage and O(dlogn) query time. In particular this yields the first algorithm for O(1)-NNS for l_∞ with subexponential storage. The preprocessing time is linear in the size of the data structure. The algorithm can be also used (after simple modifications) to output the exact nearest neighbor in time bounded bounded O(dlogn) plus the number of (4log_1+ρlog4d+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_∞ is provably hard for a version of the indexing model introduced by Hellerstein et al. (1997)