## Abstract

Let A and B be two sets of points in R^{d}, where |A| = |B| = n and the distance between them is defined by some bipartite measure dist(A, B). We study several problems in which the goal is to translate the set B, so that dist(A, B) is minimized. The main measures that we consider are (i) the diameter in two and three dimensions, that is diam(A, B) = max{d(a, b) | a ∈ A, b ∈ B}, where d(a, b) is the Euclidean distance between a and b, (ii) the uniformity in the plane, that is uni(A, B) = diam(A, B) − d(A, B), where d(A, B) = min{d(a, b) | a ∈ A, b ∈ B}, and (iii) the union width in two and three dimensions, that is union_width(A, B) = width(A ∪ B). For each of these measures we present efficient algorithms for finding a translation of B that minimizes the distance: For diameter we present near-linear-time algorithms in R^{2} and R^{3}, for uniformity we describe a roughly O(n^{9}/^{4})-time algorithm, and for union width we offer a near-linear-time algorithm in R^{2} and a quadratic-time one in R^{3}

Original language | English |
---|---|

Title of host publication | 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019 |

Editors | Rolf Niedermeier, Christophe Paul |

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

ISBN (Electronic) | 9783959771009 |

DOIs | |

State | Published - 1 Mar 2019 |

Externally published | Yes |

Event | 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019 - Berlin, Germany Duration: 13 Mar 2019 → 16 Mar 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 126 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019 |
---|---|

Country/Territory | Germany |

City | Berlin |

Period | 13/03/19 → 16/03/19 |

### Bibliographical note

Funding Information:Boris Aronov: Work on this paper by Boris Aronov was supported by NSF Grants CCF-11-17336, CCF-12-18791, and CCF-15-40656, and by grant 2014/170 from the US-Israel Binational Science Foundation. Omrit Filtser: Work on this paper by Omrit Filtser was supported by the Israel Ministry of Science, Technology & Space, and by the Lynn and William Frankel Center. Matthew J. Katz: Work on this paper by Matthew Katz was supported by grant 1884/16 from the Israel Science Foundation, and by grant 2014/170 from the US-Israel Binational Science Foundation. Khadijeh Sheikhan: Work on this paper by Khadijeh Sheikhan was supported by NSF Grant CCF-12-18791.

Funding Information:

Funding Boris Aronov: Work on this paper by Boris Aronov was supported by NSF Grants CCF-11-17336, CCF-12-18791, and CCF-15-40656, and by grant 2014/170 from the US-Israel Binational Science Foundation. Omrit Filtser: Work on this paper by Omrit Filtser was supported by the Israel Ministry of Science, Technology & Space, and by the Lynn and William Frankel Center. Matthew J. Katz: Work on this paper by Matthew Katz was supported by grant 1884/16 from the Israel Science Foundation, and by grant 2014/170 from the US-Israel Binational Science Foundation. Khadijeh Sheikhan: Work on this paper by Khadijeh Sheikhan was supported by NSF Grant CCF-12-18791.

Publisher Copyright:

© Boris Aronov, Omrit Filtser, Matthew J. Katz, and Khadijeh Sheikhan.

## Keywords

- Geometric optimization
- Minimum-width annulus
- Translation-invariant similarity measures