Small distortion and volume preserving embeddings for planar and Euclidean metrics

Abstract

The distortions and embeddings for planar and Euclidean metrics are studied. The distortion of the embedding is the maximum over pairs of the ratio of the finite metric space and the Euclidean distance between the h(u) and h(v). Improvements to Feige's and Vempala's approximation algorithms for planar and Euclidean metrics were obtained. For volume respecting embeddings, the proposed embeddings do not degrade very fast when preserving the volumes of large subsets.