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On the Number of Maximum Empty Boxes Amidst n Points

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Abstract

We revisit the following problem (along with its higher dimensional variant): Given a set S of n points inside an axis-parallel rectangle U in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in U but contains no points of S. 1. We prove that the number of maximum-area empty rectangles amidst n points in the plane is O(n log n 2^alpha(n)), where alpha(n) is the extremely slowly growing inverse of Ackermann's function. The previous best bound, O(n^2), is due to Naamad, Lee, and Hsu (1984). 2. For any d at least 3, we prove that the number of maximum-volume empty boxes amidst n points in R^d is always O(n^d) and sometimes Omega(n^floor(d/2)). This is the first superlinear lower bound derived for this problem. 3. We discuss some algorithmic aspects regarding the search for a maximum empty box in R^3. In particular, we present an algorithm that finds a (1-epsilon)-approximation of the maximum empty box amidst n points in O(epsilon^{-2} n^{5/3} log^2{n}) time.

BibTeX - Entry

@InProceedings{dumitrescu_et_al:LIPIcs:2016:5928,
author =	{Adrian Dumitrescu and Minghui Jiang},
title =	{{On the Number of Maximum Empty Boxes Amidst n Points}},
booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
pages =	{36:1--36:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-009-5},
ISSN =	{1868-8969},
year =	{2016},
volume =	{51},
editor =	{S{\'a}ndor Fekete and Anna Lubiw},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5928},
URN =		{urn:nbn:de:0030-drops-59281},
doi =		{10.4230/LIPIcs.SoCG.2016.36},
annote =	{Keywords: Maximum empty box, Davenport-Schinzel sequence, approximation algorithm, data mining.}
}

 Keywords: Maximum empty box, Davenport-Schinzel sequence, approximation algorithm, data mining. Seminar: 32nd International Symposium on Computational Geometry (SoCG 2016) Issue date: 2016 Date of publication: 2016

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