While the problem of approximate nearest neighbor search has been well-studied for Euclidean space and @math , few non-trivial algorithms are known for @math when ( @math ). In this paper, we revisit this fundamental problem and present approximate nearest-neighbor search algorithms which give the first non-trivial approximation factor guarantees in this setting.
 The author surveys algorithmic results obtained using low-distortion embeddings of metric spaces into (mostly) normed spaces. He shows that low-distortion embeddings provide a powerful and versatile toolkit for solving algorithmic problems. Their fundamental nature makes them applicable in a variety of diverse settings, while their relation to rich mathematical fields (e.g., functional analysis) ensures availability of tools for their construction. Consider a set of signals fs : 1, ..., N → [0, ..., M] appearing as a stream of tuples (i, fs (i)) in arbitrary order of i and s. We would like to devise one pass approximate algorithms for estimating various functionals on the dominant signal fmax, defined as fmax = (i, maxs fs (i)), ∀i . For example, the "worst case influence" which is the F1-norm of the dominant signal (Cormode and Muthukrishnan, 2003), general Fp-norms, and special types of distances between dominant signals. The only known previous work in this setting are the algorithms of Cormode and Muthukrishnan and Pavan and Tirtha-pura (2005) which can only estimate the F1-norm over fmax-No previous work addressed more general norms or distance estimation. In this work, we use a novel sketch, based on the properties of max-stable distributions, for these more general problems. The max-stable sketch is a significant improvement over previous alternatives in terms of simplicity of implementation, space requirements, and insertion cost, while providing similar approximation guarantees. To assert our statements, we also conduct an experimental evaluation using real datasets. We investigate the possibility of embedding an n-point metric space into a constant dimensional vector space with the maximum norm, such that the embedding is almost isometric, that is, the distortion of distances is kept arbitrarily close to 1. When the source metric is generated by any fixed norm on a finite dimensional vector space, we prove that this embedding is always possible, such that the dimension of the target space remains constant, independent of n. While this possibility has been known in the folklore, we present the first fully detailed proof, which, in addition, is significantly simpler and more transparent, then what was available before. Furthermore, our embedding can be computed in deterministic linear time in n, given oracle access to the norm.