## Abstract

In the k-dispersion problem, we need to select k nodes of a given graph so as to maximize the minimum distance between any two chosen nodes. This can be seen as a generalization of the independent set problem, where the goal is to select nodes so that the minimum distance is larger than 1. We design an optimal O(n) time algorithm for the dispersion problem on trees consisting of n nodes, thus improving the previous O(n log n) time solution from 1997. We also consider the weighted case, where the goal is to choose a set of nodes of total weight at least W. We present an O(n log2 n) algorithm improving the previous O(n log4 n) solution. Our solution builds on the search version (where we know the minimum distance λ between the chosen nodes) for which we present tight ϵ(n log n) upper and lower bounds.

Original language | English |
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Title of host publication | 25th European Symposium on Algorithms, ESA 2017 |

Editors | Christian Sohler, Christian Sohler, Kirk Pruhs |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770491 |

DOIs | |

State | Published - 1 Sep 2017 |

Event | 25th European Symposium on Algorithms, ESA 2017 - Vienna, Austria Duration: 4 Sep 2017 → 6 Sep 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 87 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 25th European Symposium on Algorithms, ESA 2017 |
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Country/Territory | Austria |

City | Vienna |

Period | 4/09/17 → 6/09/17 |

### Bibliographical note

Funding Information:∗ The research was supported in part by Israel Science Foundation grant 794/13. † The full version of this paper, containing missing proofs and supplementary figures, is available at http://arxiv.org/abs/1706.09185.

## Keywords

- Dispersion
- Dynamic programming
- K-center
- Parametric search

## ASJC Scopus subject areas

- Software