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Discrete & Computational Geometry

, Volume 19, Issue 1, pp 79–94 | Cite as

Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs

  • Y. Rabinovich
  • R. Raz

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 n 1 +O(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.

Keywords

Lower Bound General Setting Main Question Euler Characteristic Basic Technique 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag New York Inc. 1998

Authors and Affiliations

  • Y. Rabinovich
    • 1
  • R. Raz
    • 2
  1. 1.Computer Science Department, University of Toronto, Toronto, Ontario, Canada M5S 1A1 yuri@cs.toronto.eduCA
  2. 2.Department of Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel ranraz@wisdom.weizmann.ac.ilIL

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