Multiembedding of Metric Spaces

Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how large the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion as a function of the size of the metric. Using probabilistic metric embeddings, the bound on the distortion reduces to logarithmic in the size of the metric. We make a step in the direction of bypassing the lower bound on the distortion in terms of the size of the metric. We define " multiembeddings" of metric spaces, in which a point is mapped onto a set of points, while keeping the target metric of polynomial size and preserving the distortion of paths. The distortion obtained with such multiembeddings into ultrametrics is at most O(log Δ log log Δ), where Δ is the aspect ratio of the metric. In particular, for expander graphs, we are able to obtain constant distortion embeddings into trees, in contrast with the Ω(log n) lower bound for all previous notions of embeddings. We demonstrate the algorithmic application of the new embeddings for two optimization problems: group Steiner tree and metrical task systems.