2013
DOI: 10.1017/s0963548313000187
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Maximal Empty Boxes Amidst Random Points

Abstract: We show that the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube [0,1] This estimate is relevant for analyzing 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… Show more

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Cited by 6 publications
(2 citation statements)
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“…The problem of investigating empty substructures in random point sets seems to have received far less attention; we could only find two related results in the literature, other than the already mentioned [2]. In [6], Dumitrescu and Jiang found the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube; and, very recently, Fabila-Monroy et al found the expected number of empty convex (and also non-convex) four-gons with vertices in a finite set randomly chosen from a convex set [9].…”
Section: Introductionmentioning
confidence: 92%
“…The problem of investigating empty substructures in random point sets seems to have received far less attention; we could only find two related results in the literature, other than the already mentioned [2]. In [6], Dumitrescu and Jiang found the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube; and, very recently, Fabila-Monroy et al found the expected number of empty convex (and also non-convex) four-gons with vertices in a finite set randomly chosen from a convex set [9].…”
Section: Introductionmentioning
confidence: 92%
“…This quantity was proven to be essential for various numerical problems including optimization [13], approximation of high-dimensional rank-1 tensors [2,15] and, very recently, in the study of Marcinkiewicz-type discretization theorems [22,23,24]. Moreover, algorithms for finding such a box with maximal volume, with a motivation coming from information theory, are also of some interest, see [4,5,11] and references therein. In all these contexts, it is desirable to have tight bounds Date: September 3, 2018.…”
mentioning
confidence: 99%