Low-Distortion Embeddings of Trees

Low distortion Eu lidean embeddings of trees

Combinatorial theorems about embedding trees on the real line

We consider the combinatorial problem of embedding a tree metric into the real line with low distortion. For two special families of trees — the family of complete binary trees and the family of subdivided stars — we provide embeddings whose distortion is provably optimal, up to a constant factor. We also prove that the optimal distortion of a linear embedding of a tree can be arbitrarily low o...

Multiembedding of Metric Spaces

Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding’s distortion is. For example, embedding finite metric space into trees may require linear distortion as a function of its size. Using probabilistic metric embeddings, the bound on the distortion reduces to logarithmic in the size. We make a step in the direc...

Steiner Shallow-Light Trees are Exponentially Better than Spanning Ones

The power of Steiner points was studied in a number of different settings in the context of metric embeddings. Perhaps most notably in the context of probabilistic tree embeddings Bartal and Fakcharoenphol et al. [8, 9, 21] used Steiner points to devise near-optimal constructions of such embeddings. However, Konjevod et al. [24] and Gupta [22] demonstrated that Steiner points do not help in thi...

Distortion of embeddings of binary trees into diamond graphs

Diamond graphs and binary trees are important examples in the theory of metric embeddings and also in the theory of metric characterizations of Banach spaces. Some results for these families of graphs are parallel to each other, for example superreflexivity of Banach spaces can be characterized both in terms of binary trees (Bourgain, 1986) and diamond graphs (Johnson-Schechtman, 2009). In this...