We show that the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube [0,1] d in ℝ d is \((1 \pm o(1))\allowbreak \frac{(2d-2)!}{(d-1)!} \, n \ln^{d-1} n\), if d is fixed. This estimate is relevant for analyzing the performance of any exact algorithm for computing the largest empty axis-parallel box amidst n points in a given axis-parallel box R, that proceeds by examining all maximal empty boxes. While the Θ(n log d − 1 n) bound has been claimed for d = 3 for more than ten years by now, and has been recently used for all d ≥ 3 in the analysis of algorithms for computing the largest empty box, it did not rely on a valid proof. Here we present the first valid proof for the Θ(n log d − 1 n) bound; only an O(n log d − 1 n) bound was previously proved.


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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Minghui Jiang
    • 2
  1. 1.Department of Computer ScienceUniversity of Wisconsin–MilwaukeeUSA
  2. 2.Department of Computer ScienceUtah State UniversityLoganUSA

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