The mono- and bichromatic empty rectangle and square problems in all dimensions

Authors:
Jonathan Backer;J. Mark Keil
Affiliations:
Department of Computer Science, University of Saskatchewan, Canada;Department of Computer Science, University of Saskatchewan, Canada
Venue:
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Year:
2010

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Abstract

The maximum empty rectangle problem is as follows: Given a set of red points in ℝd and an axis-aligned hyperrectangle B, find an axis-aligned hyperrectangle R of greatest volume that is contained in B and contains no red points. In addition to this problem, we also consider three natural variants: where we find a hypercube instead of a hyperrectangle, where we try to contain as many blue points as possible instead of maximising volume, and where we do both. Combining the results of this paper with previous results, we now know that all four of these problems (a) are NP-complete if d is part of the input, (b) have polynomial-time sweep-plane solutions for any fixed d≥3, and (c) have near linear time solutions in two dimensions.