Embedding Tree Metrics into Low-Dimensional Euclidean Spaces

We consider embedding metrics induced by trees into Euclidean spaces with a restricted number of dimensions. We show that any weighted tree T with n vertices and L leaves can be embedded into d -dimensional Euclidean space with Õ (L 1/(d-1) ) distortion. Furthermore, we exhibit an embedding with almost the same distortion which can be computed efficiently. This distortion substantially improves the previous best upper bound of \tilde O (n 2/d ) and almost matches the best known lower bound of Ω(L 1/d ) .