Lower bounds on the distortion of embedding finite metric spaces in graphs

Y. Rabinovich, R. Raz

Research output: Contribution to journalArticle

63 Scopus citations

Abstract

The main question discussed in this paper is how well a finite metric space of size n can be embedded into a graph with certain topological restrictions. The existing constructions of graph spanners imply that any n-point metric space can be represented by a (weighted) graph with n vertices and n1+0(1/r) edges, with distances distorted by at most r. We show that this tradeoff between the number of edges and the distortion cannot be improved, and that it holds in a much more general setting. The main technical lemma claims that the metric space induced by an unweighted graph H of girth g cannot be embedded in a graph G (even if it is weighted) of smaller Euler characteristic, with distortion less than g/4 - 3/2. In the special case when | V(G)\ = | V(H)| and G has strictly less edges than H, an improved bound of g/3 - 1 is shown. In addition, we discuss the case χ(G) < χ(H) - 1, as well as some interesting higher-dimensional analogues. The proofs employ basic techniques of algebraic topology.

Original languageEnglish (US)
Pages (from-to)79-94
Number of pages16
JournalDiscrete and Computational Geometry
Volume19
Issue number1
DOIs
StatePublished - Jan 1998
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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