Embedding tree metrics into low-dimensional Euclidean spaces

Abstract

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 (L1/(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 Õ (n2/d) and almost matches the best known lower bound of Ω (L1/d).