The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (σ-immanent description). Constructing the geometry, one does not use topology and topological properties. For instance, the straight, passing through points A and B, is defined as a set of such points R that the area S(A,B,R) of triangle ABR vanishes. The triangle ar...

We study approximation algorithms for k-median clustering. We obtain small coresets for k-median clustering in metric spaces as well as in Euclidean spaces. Specifically, in IR, those coresets are of size with only polynomial dependency on d. This leads to a (1 + ε)-approximation algorithm for kmedian clustering in IR, with running time O(ndk + 2 O(1) dn), for any σ > 0. This is an improvement ...

We study the problem of how well a tree metric is able to preserve the sum of pairwisedistances of an arbitrary metric. This problem is closely related to low-stretch metricembeddings and is interesting by its own flavor from the line of research proposed in theliterature.As the structure of a tree imposes great constraints on the pairwise distances, any embed-ding of a ...