### 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

**Bipartite diameter and other measures under translation.** / Aronov, Boris; Filtser, Omrit; Katz, Matthew J.; Sheikhan, Khadijeh.

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

*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, 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, Berlin, Germany, 3/13/19. https://doi.org/10.4230/LIPIcs.STACS.2019.8

}

TY - GEN

T1 - Bipartite diameter and other measures under translation

AU - Aronov, Boris

AU - Filtser, Omrit

AU - Katz, Matthew J.

AU - Sheikhan, Khadijeh

PY - 2019/3/1

Y1 - 2019/3/1

N2 - Let A and B be two sets of points in Rd, 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 R2 and R3, for uniformity we describe a roughly O(n9/4)-time algorithm, and for union width we offer a near-linear-time algorithm in R2 and a quadratic-time one in R3

AB - Let A and B be two sets of points in Rd, 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 R2 and R3, for uniformity we describe a roughly O(n9/4)-time algorithm, and for union width we offer a near-linear-time algorithm in R2 and a quadratic-time one in R3

KW - Geometric optimization

KW - Minimum-width annulus

KW - Translation-invariant similarity measures

UR - http://www.scopus.com/inward/record.url?scp=85074924983&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85074924983&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.STACS.2019.8

DO - 10.4230/LIPIcs.STACS.2019.8

M3 - Conference contribution

AN - SCOPUS:85074924983

T3 - Leibniz International Proceedings in Informatics, LIPIcs

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

A2 - Niedermeier, Rolf

A2 - Paul, Christophe

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -