## Abstract

We introduce a new notion of embedding, called minimum-relaxation ordinal embedding, parallel to the standard notion of minimum-distortion (metric) embedding. In an ordinal embedding, it is the relative order between pairs of distances, and not the distances themselves, that must be preserved as much as possible. The (multiplicative) relaxation of an ordinal embedding is the maximum ratio between two distances whose relative order is inverted by the embedding. We develop several worst-case bounds and approximation algorithms on ordinal embedding. In particular, we establish that ordinal embedding has many qualitative differences from metric embedding, and capture the ordinal behavior of ultrametrics and shortest-path metrics of unweighted trees.

Original language | English (US) |
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Pages | 650-659 |

Number of pages | 10 |

State | Published - Jul 1 2005 |

Externally published | Yes |

Event | Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms - Vancouver, BC, United States Duration: Jan 23 2005 → Jan 25 2005 |

### Other

Other | Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country/Territory | United States |

City | Vancouver, BC |

Period | 1/23/05 → 1/25/05 |

## All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)