### Abstract

Let (X, d_{x}) be an n-point metric space. We show that there exists a distribution & over non-contractive embeddings into trees f : X → T such that for every x ∈ X, E_{D} [max _{v∈X\(x)} d_{T}(f(x), f(y))/d_{X}(x, y)] < C(log n)_{2} where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.

Original language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 10th International Workshop, APPROX 2007 and 11th International Workshop, RANDOM 2007, Proceedings |

Publisher | Springer Verlag |

Pages | 242-256 |

Number of pages | 15 |

ISBN (Print) | 9783540742074 |

DOIs | |

State | Published - 2007 |

Event | 10th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2007 and 11th International Workshop on Randomization and Computation, RANDOM 2007 - Princeton, NJ, United States Duration: Aug 20 2007 → Aug 22 2007 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4627 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 10th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2007 and 11th International Workshop on Randomization and Computation, RANDOM 2007 |
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Country | United States |

City | Princeton, NJ |

Period | 8/20/07 → 8/22/07 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

Mendel, M., & Naor, A. (2007). Maximum gradient embeddings and monotone clustering. In

*Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 10th International Workshop, APPROX 2007 and 11th International Workshop, RANDOM 2007, Proceedings*(pp. 242-256). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4627 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-540-74208-1_18