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Discrete & Computational Geometry

, Volume 59, Issue 3, pp 742–756 | Cite as

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. (I)

    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 et al. (Discrete Appl Math 8(3):267–277, 1984).

     
  2. (II)

    For any \(d \ge 3\), we prove that the number of maximum-volume empty boxes amidst n points in \(\mathbb {R}^d\) is always \(O(n^d)\) and sometimes \(\Omega (n^{\lfloor d/2 \rfloor })\). This is the first superlinear lower bound derived for this problem.

     
  3. (III)

    We discuss some algorithmic aspects regarding the search for a maximum empty box in \(\mathbb {R}^3\). In particular, we present an algorithm that finds a \((1-\varepsilon )\)-approximation of the maximum empty box amidst n points in \(O(\varepsilon ^{-2} n^{5/3} \log ^2{n})\) time.

     

Keywords

Maximum empty box Davenport–Schinzel sequence Approximation algorithm Data mining 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Wisconsin–MilwaukeeMilwaukeeUSA
  2. 2.Department of Computer ScienceUtah State UniversityLoganUSA

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