On the Number of Maximum Empty Boxes Amidst n Points

Authors Adrian Dumitrescu, Minghui Jiang

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Adrian Dumitrescu
Minghui Jiang

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Adrian Dumitrescu and Minghui Jiang. On the Number of Maximum Empty Boxes Amidst n Points. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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.
  • Maximum empty box
  • Davenport-Schinzel sequence
  • approximation algorithm
  • data mining.


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