### 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) |
---|---|

Title of host publication | Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 10th International Workshop, APPROX 2007 and 11th International Workshop, RANDOM 2007, Proceedings |

Pages | 242-256 |

Number of pages | 15 |

Volume | 4627 LNCS |

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) |
---|---|

Volume | 4627 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

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

Country | United States |

City | Princeton, NJ |

Period | 8/20/07 → 8/22/07 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

*Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 10th International Workshop, APPROX 2007 and 11th International Workshop, RANDOM 2007, Proceedings*(Vol. 4627 LNCS, pp. 242-256). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4627 LNCS).

**Maximum gradient embeddings and monotone clustering.** / Mendel, Manor; Naor, Assaf.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 10th International Workshop, APPROX 2007 and 11th International Workshop, RANDOM 2007, Proceedings.*vol. 4627 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4627 LNCS, pp. 242-256, 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, 8/20/07.

}

TY - GEN

T1 - Maximum gradient embeddings and monotone clustering

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=38049060509&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38049060509&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9783540742074

VL - 4627 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 242

EP - 256

BT - Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 10th International Workshop, APPROX 2007 and 11th International Workshop, RANDOM 2007, Proceedings

ER -