## Abstract

We introduce and study the notion of the average distortion of a nonexpanding embedding of one metric space into another. Less sensitive than the multiplicative metric distortion, the average distortion captures well the global picture and, overall, is a quite interesting new measure of metric proximity, related to the concentration of measure phenomenon. The paper mostly deals with embeddings into the real line with a low (as much as it is possible) average distortion. Our main technical contribution is that the shortest-path metrics of special (e.g., planar, bounded treewidth, etc.) undirected weighted graphs can be embedded into the line with constant average distortion. This has implications, e.g., on the value of the MinCut-MaxFlow gap in uniform-demand multicommodity flows on such graphs.

Original language | English |
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Pages (from-to) | 720-733 |

Number of pages | 14 |

Journal | Discrete and Computational Geometry |

Volume | 39 |

Issue number | 4 |

DOIs | |

State | Published - Jun 2008 |

### Bibliographical note

Funding Information:Supported in part by a grant ISF-247-02-10.5.

## Keywords

- Average distortion
- Metric embeddings

## ASJC Scopus subject areas

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