When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2019.8
URN: urn:nbn:de:0030-drops-102476
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10247/
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### Bipartite Diameter and Other Measures Under Translation

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### 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 in A, b in 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 in A, b in B}, and (iii) the union width in two and three dimensions, that is union_width(A,B) = width(A cup 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.

### BibTeX - Entry

```@InProceedings{aronov_et_al:LIPIcs:2019:10247,
author =	{Boris Aronov and Omrit Filtser and Matthew J. Katz and Khadijeh Sheikhan},
title =	{{Bipartite Diameter and Other Measures Under Translation}},
booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
pages =	{8:1--8:14},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-100-9},
ISSN =	{1868-8969},
year =	{2019},
volume =	{126},
editor =	{Rolf Niedermeier and Christophe Paul},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},