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Maximal Empty Boxes Amidst Random Points

Published online by Cambridge University Press:  14 May 2013

ADRIAN DUMITRESCU  and
MINGHUI JIANG
ADRIAN DUMITRESCU
Affiliation:
Department of Computer Science, University of Wisconsin–Milwaukee, USA (e-mail: [email protected])
MINGHUI JIANG
Affiliation:
Department of Computer Science, Utah State University, Logan, USA (e-mail: [email protected])
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Abstract

We show that the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube [0,1]d in $\mathbb{R}^d$ is (1 ± o(1)) $\frac{(2d-2)!}{(d-1)!}$n lnd−1n, if d is fixed. This estimate is relevant to analysis of the performance of exact algorithms for computing the largest empty axis-parallel box amidst n given points in an axis-parallel box R, especially the algorithms that proceed by examining all maximal empty boxes. Our method for bounding the expected number of maximal empty boxes also shows that the expected number of maximal empty orthants determined by n random points in $\mathbb{R}^d$ is (1 ± o(1)) lnd−1n, if d is fixed. This estimate is related to the expected number of maximal (or minimal) points amidst random points, and has application to algorithms for coloured orthogonal range counting.

Keywords

Primary 52C45Secondary 68U0568W25
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 4 , July 2013 , pp. 477 - 498
Copyright
Copyright © Cambridge University Press 2013 

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