Abstract
We give the first efficient (1−ε)-approximation algorithm for the following problem: Given an axis-parallel d-dimensional box R in ℝd containing n points, compute a maximum-volume empty axis-parallel d-dimensional box contained in R. The minimum of this quantity over all such point sets is of the order \(\Theta (\frac {1}{n} )\). Our algorithm finds an empty axis-aligned box whose volume is at least (1−ε) of the maximum in O((8edε −2)d⋅nlogd n) time. No previous efficient exact or approximation algorithms were known for this problem for d≥4. As the problem has been recently shown to be NP-hard in arbitrarily high dimensions (i.e., when d is part of the input), the existence of an efficient exact algorithm is unlikely.
We also present a (1−ε)-approximation algorithm that, given an axis-parallel d-dimensional cube R in ℝd containing n points, computes a maximum-volume empty axis-parallel hypercube contained in R. The minimum of this quantity over all such point sets is also shown to be of the order \(\Theta (\frac{1}{n} )\). A faster (1−ε)-approximation algorithm, with a milder dependence on d in the running time, is obtained in this case.
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Notes
Let ℓ(v) and ℓ′(v) denote the two orthogonal lines incident to a vertex v of U. It is easy to see that there exists a vertex v of U such that when ℓ(v) and ℓ′(v) are translated towards the interior of the square, they hit two distinct points out of the six contained in U. By balancing the areas of the two rectangles swept by these two lines, say R 1 and R 2, with the area of the largest empty sub-rectangle inside the rectangle U∖(R 1∪R 2) as guaranteed by Lemma 2 and Observation 1, we get that A(6)≥x, where \(x=3 -2\sqrt{2}\) is the solution of the quadratic equation x=(1−x)2/4.
The argument we use here is similar to that used for bounding the geometric discrepancy of the van der Corput set of points.
For d=2 volume is replaced by area throughout this proof.
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The authors would like to thank the anonymous reviewers for careful reading and thoughtful suggestions.
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A. Dumitrescu was supported in part by NSF CAREER grant CCF-0444188 and NSF grant DMS-1001667. Part of the research by this author was done at Ecole Polytechnique Fédérale de Lausanne.
M. Jiang was supported in part by NSF grant DBI-0743670.
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Dumitrescu, A., Jiang, M. On the Largest Empty Axis-Parallel Box Amidst n Points. Algorithmica 66, 225–248 (2013). https://doi.org/10.1007/s00453-012-9635-5
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DOI: https://doi.org/10.1007/s00453-012-9635-5