## Abstract

This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of ∈, with the guarantee that for each ∈ the distortion of a fraction 1 - ∈ of all pairs is bounded accordingly. Such a bound implies, in particular, that the average distortion and ℓ_{q}-distortions are small. Specifically, our embeddings have constant average distortion and O(√log n)ℓ_{2}-distortion. This follows from the following results: we prove that any metric space embeds into an ultrametric with scaling distortion O(√1/∈). For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion O(√1/∈). These bounds are tight even for embedding in arbitrary trees. For probabilistic embedding into spanning trees we prove a scaling distortion of Õ (log^{2}(1/∈)), which implies constant ℓ_{q}-distortion for every fixed q < ∞.

Original language | American English |
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Title of host publication | Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 |

Publisher | Association for Computing Machinery |

Pages | 502-511 |

Number of pages | 10 |

ISBN (Electronic) | 9780898716245 |

State | Published - 2007 |

Event | 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 - New Orleans, United States Duration: 7 Jan 2007 → 9 Jan 2007 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 07-09-January-2007 |

### Conference

Conference | 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 |
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Country/Territory | United States |

City | New Orleans |

Period | 7/01/07 → 9/01/07 |

### Bibliographical note

Publisher Copyright:Copyright © 2007 by the Association for Computing Machinery, Inc. and the Society for Industrial and Applied Mathematics.