### Abstract

In online clustering problems, the classification of points into sets (called clusters) is done in an online fashion. Points arrive one by one at arbitrary locations, to be assigned to clusters at the time of arrival. A point can be assigned to an existing cluster, or a new cluster can be opened for it. We study a one dimensional variant on a line, where there is no restriction on the length of a cluster, and the cost of a cluster is the sum of a fixed set-up cost and its diameter. The goal is to minimize the sum of costs of the clusters used by the algorithm. We study several variants, all maintaining the essential property that a point which was assigned to a given cluster must remain assigned to this cluster, and clusters cannot be merged. In the strict variant, the diameter and the exact location of the cluster must be fixed when it is initialized. In the flexible variant, the algorithm can shift the cluster or expand it, as long as it contains all points assigned to it. In an intermediate model, the diameter is fixed in advance while the exact location can be modified. We give tight bounds on the competitive ratio of any online algorithm in each of these variants. In addition, for each one of the models, we consider also the semi-online case, where points are presented sorted by their location.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 282-293 |

Number of pages | 12 |

Volume | 6281 LNCS |

DOIs | |

Publication status | Published - 2010 |

Event | 35th International Symposium on Mathematical Foundations of Computer Science, MFCS 2010 - Brno, Czech Republic Duration: Aug 23 2010 → Aug 27 2010 |

### 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 | 6281 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 35th International Symposium on Mathematical Foundations of Computer Science, MFCS 2010 |
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Country | Czech Republic |

City | Brno |

Period | 8/23/10 → 8/27/10 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 6281 LNCS, pp. 282-293). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6281 LNCS). https://doi.org/10.1007/978-3-642-15155-2_26

**Online clustering with variable sized clusters.** / Csirik, J.; Epstein, Leah; Imreh, Csanád; Levin, Asaf.

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

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 6281 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6281 LNCS, pp. 282-293, 35th International Symposium on Mathematical Foundations of Computer Science, MFCS 2010, Brno, Czech Republic, 8/23/10. https://doi.org/10.1007/978-3-642-15155-2_26

}

TY - GEN

T1 - Online clustering with variable sized clusters

AU - Csirik, J.

AU - Epstein, Leah

AU - Imreh, Csanád

AU - Levin, Asaf

PY - 2010

Y1 - 2010

N2 - In online clustering problems, the classification of points into sets (called clusters) is done in an online fashion. Points arrive one by one at arbitrary locations, to be assigned to clusters at the time of arrival. A point can be assigned to an existing cluster, or a new cluster can be opened for it. We study a one dimensional variant on a line, where there is no restriction on the length of a cluster, and the cost of a cluster is the sum of a fixed set-up cost and its diameter. The goal is to minimize the sum of costs of the clusters used by the algorithm. We study several variants, all maintaining the essential property that a point which was assigned to a given cluster must remain assigned to this cluster, and clusters cannot be merged. In the strict variant, the diameter and the exact location of the cluster must be fixed when it is initialized. In the flexible variant, the algorithm can shift the cluster or expand it, as long as it contains all points assigned to it. In an intermediate model, the diameter is fixed in advance while the exact location can be modified. We give tight bounds on the competitive ratio of any online algorithm in each of these variants. In addition, for each one of the models, we consider also the semi-online case, where points are presented sorted by their location.

AB - In online clustering problems, the classification of points into sets (called clusters) is done in an online fashion. Points arrive one by one at arbitrary locations, to be assigned to clusters at the time of arrival. A point can be assigned to an existing cluster, or a new cluster can be opened for it. We study a one dimensional variant on a line, where there is no restriction on the length of a cluster, and the cost of a cluster is the sum of a fixed set-up cost and its diameter. The goal is to minimize the sum of costs of the clusters used by the algorithm. We study several variants, all maintaining the essential property that a point which was assigned to a given cluster must remain assigned to this cluster, and clusters cannot be merged. In the strict variant, the diameter and the exact location of the cluster must be fixed when it is initialized. In the flexible variant, the algorithm can shift the cluster or expand it, as long as it contains all points assigned to it. In an intermediate model, the diameter is fixed in advance while the exact location can be modified. We give tight bounds on the competitive ratio of any online algorithm in each of these variants. In addition, for each one of the models, we consider also the semi-online case, where points are presented sorted by their location.

UR - http://www.scopus.com/inward/record.url?scp=78349254592&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78349254592&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-15155-2_26

DO - 10.1007/978-3-642-15155-2_26

M3 - Conference contribution

SN - 364215154X

SN - 9783642151545

VL - 6281 LNCS

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

SP - 282

EP - 293

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

ER -