Approximation Algorithms for Low-Distortion Embeddings into Low-Dimensional Spaces

We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the two-dimensional plane. Among other results, we give an $O\left(\sqrt{n}\right)$-approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph, that minimizes the (standard) multiplicative distortion. We give an improved $\tilde{O}\left(n^{1/3}\right)$ approximation for the case of metrics generated by unweighted trees. This is the first result of this type.