### 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 (US) |
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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 - Mar 1 2019 |

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

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 126 |

ISSN (Print) | 1868-8969 |

### Conference

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

City | Berlin |

Period | 3/13/19 → 3/16/19 |

### Keywords

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

### ASJC Scopus subject areas

- Software

## Cite this

*36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019*[8] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 126). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.STACS.2019.8