Abstract
The maximum empty rectangle problem is as follows: Given a set of red points in ād 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 dāā„ā3, and (c) have near linear time solutions in two dimensions.
This research is supported by the Natural Sciences and Engineering Research Council (NSERC).
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References
Naamad, A., Lee, D., Hsu, W.: Maximum empty rectangle problem. Discrete Appl. Math.Ā 8(3), 267ā277 (1984)
Chazelle, B., Drysdale, R., Lee, D.: Computing the largest empty rectangle. SIAM J. Comput.Ā 15, 300 (1986)
Aggarwal, A., Suri, S.: Fast algorithms for computing the largest empty rectangle. In: Symposium on Computational Geometry, pp. 278ā290 (1987)
Nandy, S., Bhattacharya, B.: Maximal empty cuboids among points and blocks. Computers and Mathematics with ApplicationsĀ 36(3), 11ā20 (1998)
Orlowski, M.: A new algorithm for the largest empty rectangle problem. AlgorithmicaĀ 5(1), 65ā73 (1990)
Datta, A., Soundaralakshmi, S.: An efficient algorithm for computing the maximum empty rectangle in three dimensions. Inform. SciencesĀ 128(1-2), 43ā65 (2000)
Eckstein, J., Hammer, P., Liu, Y., Nediak, M., Simeone, B.: The maximum box problem and its application to data analysis. Comput. Optim. Appl.Ā 23(3), 285ā298 (2002)
Aronov, B., Har-Peled, S.: On Approximating the Depth and Related Problems. SIAM J. Comput.Ā 38(3), 899ā921 (2008)
Liu, Y., Nediak, M.: Planar case of the maximum box and related problems. In: CCCG, pp. 14ā18 (2003)
Edmonds, J., Gryz, J., Liang, D., Miller, R.J.: Mining for empty spaces in large data sets. Theor. Comput. Sci.Ā 296(3), 435ā452 (2003)
Dobkin, D., Gunopulos, D., Maass, W.: Computing the maximum bichromatic discrepancy, with applications to computer graphics and machine learning. J. Comput. Syst. Sci.Ā 52(3), 453ā470 (1996)
Frederickson, G.N., Johnson, D.B.: Generalized selection and ranking: Sorted matrices. SIAM J. Comput.Ā 13(1), 14ā30 (1984)
Overmars, M., Yap, C.: New upper bounds in Kleeās measure problem. SIAM J. Comput.Ā 20(6), 1034ā1045 (1991)
Boissonnat, J.D., Sharir, M., Tagansky, B., Yvinec, M.: Voronoi diagrams in higher dimensions under certain polyhedral distance functions. Discrete & Computational GeometryĀ 19(4), 485ā519 (1998)
Kaplan, H., Rubin, N., Sharir, M., Verbin, E.: Counting colors in boxes. In: Bansal, N., Pruhs, K., Stein, C. (eds.) SODA, pp. 785ā794. SIAM, Philadelphia (2007)
Lee, D.T., Wong, C.K.: Voronoi diagrams in l 1 (l āāā) metrics with 2-dimensional storage applications. SIAM J. Comput.Ā 9(1), 200ā211 (1980)
Aggarwal, A., Klawe, M., Moran, S., Shor, P., Wilber, R.: Geometric applications of a matrix-searching algorithm. AlgorithmicaĀ 2(1), 195ā208 (1987)
McKenna, M., OāRourke, J., Suri, S.: Finding the largest rectangle in an orthogonal polygon. In: Allerton Conference, pp. 486ā495 (1985)
Dumitrescu, A., Jiang, M.: On the largest empty axis-parallel box amidst n points. arxiv.org reference arXiv:0909.3127 (2009) Submitted for publication
Agarwal, P.K., Kaplan, H., Sharir, M.: Computing the volume of the union of cubes. In: Erickson, J. (ed.) Symposium on Computational Geometry, pp. 294ā301. ACM, New York (2007)
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Backer, J., Keil, J.M. (2010). The Mono- and Bichromatic Empty Rectangle and Square Problems in All Dimensions. In: LĆ³pez-Ortiz, A. (eds) LATIN 2010: Theoretical Informatics. LATIN 2010. Lecture Notes in Computer Science, vol 6034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12200-2_3
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