Trees and Euclidean metrics

In order to study a finite metric space (X,d), one oflen aeeka first an approximation in the form of a metric that is induced from a norm. The quality of such an approximation is quantified by the distortion of the corrcaponda’ng embedding, i.e., the Lipschitz constant of tlbe mapping. We concentrate on embedding into Euclidean spaces, and introduce the notation cz(X,d) the least distortion with which (X,d) may be embedded in any Euclidean apace. It is known that if (X,d) has n points, then cz(X, d) ,< O(log n) and the bound is tight. Let T be a tree with n vertices, and d be the metric induced by it. We show that cz(T,d) 5 O(Ioglogn), that i3 we provide an embedding f of T’s vertices into h+&lean apace, such that d(w) S IIf@) fb)II I CWw ~x,Y) for some constant C. This embedding can be computed e.& cfently.