Finite Metric Spaces & Their Embeddings: Introduction and Basic Tools

Definition of (semi) metric. CS motivation. Finite metric spaces arise naturally in combinatorial objects, and algo-rithmic questions. For example, as the shortest path metrics on graphs. We will also see less obvious connections. Properties of finite metrics. The following properties have been investigated: Dimension , extendability of Lipschitz and Hölder functions, decomposability, Inequalities satisfied by the metric, short representations, additive distortion of embedding, (multiplicative) distortion of embeddings. We will focus on the last property. Embedding. A mapping f : (M, ρ) → (H, ν) of a metric space M into a host metric space H, that (hopefully) preserves the geometry of M (usually distances). The distortion of f is denoted by dist(f). C is a scaling factor. Another way to define the distortion: The Lipschitz constant of a mapping f : (M, ρ) → (H, ν) is f Lip = max x,y∈M ρ(x,y)>0 ν(f (x), f (y)) ρ(x, y). Then dist(f) = f Lip · f −1 Lip .