Computational metric embeddings

We study the problem of computing a low-distortion embedding between two metric spaces. More precisely given an input metric space M we are interested in computing in polynomial time an embedding into a host space M ′ with minimum multiplicative distortion. This problem arises naturally in many applications, including geometric optimization, visualization, multi-dimensional scaling, network spanners, and the computation of phylogenetic trees. We focus on the case where the host space is either a euclidean space of constant dimension such as the line and the plane, or a graph metric of simple topological structure such as a tree. For Euclidean spaces, we present the following upper bounds. We give an approximation algorithm that, given a metric space that embeds into R with distortion c, computes an embedding with distortion c∆ (∆ denotes the ratio of the maximum over the minimum distance). For higher-dimensional spaces, we obtain an algorithm which, for any fixed d ≥ 2, given an ultrametric that embeds into R with distortion c, computes an embedding with distortion c. We also present an algorithm achieving distortion c log∆ for the same problem. We complement the above upper bounds by proving hardness of computing optimal, or near-optimal embeddings. When the input space is an ultrametric, we show that it is NP-hard to compute an optimal embedding into R under the l∞ norm. Moreover, we prove that for any fixed d ≥ 2, it is NP-hard to approximate the minimum distortion embedding of an n-point metric space into R within a factor of Ω(n). Finally, we consider the problem of embedding into tree metrics. We give a O(1)approximation algorithm for the case where the input is the shortest-path metric of an unweighted graph. For general metric spaces, we present an algorithm which, given an n-point metic that embeds into a tree with distortion c, computes an embedding with distortion (c logn) √ . By composing this algorithm with an algorithm for embedding trees into R, we obtain an improved algorithm for embedding general metric spaces into R. Thesis Supervisor: Piotr Indyk