Online Embeddings

Abstract

We initiate the study of on-line metric embeddings. In such an embedding we are given a sequence of n points X = x [subscript 1],...,x [subscript n] one by one, from a metric space M = (X,D). Our goal is to compute a low-distortion embedding of M into some host space, which has to be constructed in an on-line fashion, so that the image of each x i depends only on x [subscript 1],...,x [subscript i] . We prove several results translating existing embeddings to the on-line setting, for the case of embedding into ℓ [subscript p] spaces, and into distributions over ultrametrics.