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On the Number of Maximum Empty Boxes Amidst n Points

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.

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Notes

  1. A weaker bound with \(b=3\) was inadvertently labeled as an improvement over this bound in [18].

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Correspondence to Adrian Dumitrescu.

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Dumitrescu, A., Jiang, M. On the Number of Maximum Empty Boxes Amidst n Points. Discrete Comput Geom 59, 742–756 (2018). https://doi.org/10.1007/s00454-017-9871-1

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  • DOI: https://doi.org/10.1007/s00454-017-9871-1

Keywords

  • Maximum empty box
  • Davenport–Schinzel sequence
  • Approximation algorithm
  • Data mining