Abstract
We show that the expected number of maximal empty axis-parallel boxes amidst n random points in the unit hypercube [0,1]d in ℝd is \((1 \pm o(1))\allowbreak \frac{(2d-2)!}{(d-1)!} \, n \ln^{d-1} n\), if d is fixed. This estimate is relevant for analyzing the performance of any exact algorithm for computing the largest empty axis-parallel box amidst n points in a given axis-parallel box R, that proceeds by examining all maximal empty boxes. While the Θ(n logd − 1 n) bound has been claimed for d = 3 for more than ten years by now, and has been recently used for all d ≥ 3 in the analysis of algorithms for computing the largest empty box, it did not rely on a valid proof. Here we present the first valid proof for the Θ(n logd − 1 n) bound; only an O(n logd − 1 n) bound was previously proved.
Due to space constraints, we omit the proofs of some lemmas in this extended abstract.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aggarwal, A., Suri, S.: Fast algorithms for computing the largest empty rectangle. In: Proceedings of the 3rd Annual Symposium on Computational Geometry, pp. 278–290 (1987)
Atallah, M., Frederickson, G.: A note on finding the maximum empty rectangle. Discrete Applied Mathematics 13, 87–91 (1986)
Atallah, M., Kosaraju, S.R.: An efficient algorithm for maxdominance, with applications. Algorithmica 4, 221–236 (1989)
Backer, J., Keil, M.: The bichromatic rectangle problem in high dimensions. In: Proceedings of the 21st Canadian Conference on Computational Geometry, pp. 157–160 (2009)
Backer, J., Keil, J.M.: The Mono- and Bichromatic Empty Rectangle and Square Problems in All Dimensions. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 14–25. Springer, Heidelberg (2010)
Bentley, J.L., Kung, H.T., Schkolnick, M., Thompson, C.D.: On the average number of maxima in a set of vectors and applications. Journal of the ACM 25, 536–543 (1978)
Chazelle, B., Drysdale, R., Lee, D.T.: Computing the largest empty rectangle. SIAM Journal on Computing 15, 300–315 (1986)
Chuan-Chong, C., Khee-Meng, K.: Principles and Techniques in Combinatorics. World Scientific, Singapore (1996)
Datta, A.: Efficient algorithms for the largest empty rectangle problem. Information Sciences 64, 121–141 (1992)
Datta, A., Soundaralakshmi, S.: An efficient algorithm for computing the maximum empty rectangle in three dimensions. Information Sciences 128, 43–65 (2000)
Dumitrescu, A., Jiang, M.: On the largest empty axis-parallel box amidst n points. Algorithmica (2012), doi:10.1007/s00453-012-9635-5
Edmonds, J., Gryz, J., Liang, D., Miller, R.: Mining for empty spaces in large data sets. Theoretical Computer Science 296, 435–452 (2003)
Giannopoulos, P., Knauer, C., Wahlström, M., Werner, D.: Hardness of discrepancy computation and ε-net verification in high dimension. Journal of Complexity (2011), doi:10.1016/j.jco.2011.09.001
Kaplan, H., Rubin, N., Sharir, M., Verbin, E.: Efficient colored orthogonal range counting. SIAM Journal on Computing 38, 982–1011 (2008)
Klein, R.: Direct dominance of points. International Journal of Computer Mathematics 19, 225–244 (1986)
Kudryavtsev, L.D.: The method of undetermined coefficients. In: Hazewinkel, M. (ed.) Encyclopaedia of Mathematics. Springer (2001)
Kurosh, A.: Higher Algebra. Mir Publishers, Moscow (1975)
Marx, D.: Parameterized complexity and approximation algorithms. Computer Journal 51, 60–78 (2008)
McKenna, M., O’Rourke, J., Suri, S.: Finding the largest rectangle in an orthogonal polygon. In: Proceedings of the 23rd Annual Allerton Conference on Communication, Control and Computing, Urbana-Champaign, Illinois (October 1985)
Naamad, A., Lee, D.T., Hsu, W.-L.: On the maximum empty rectangle problem. Discrete Applied Mathematics 8, 267–277 (1984)
Orlowski, M.: A new algorithm for the largest empty rectangle problem. Algorithmica 5, 65–73 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dumitrescu, A., Jiang, M. (2012). Maximal Empty Boxes Amidst Random Points. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_45
Download citation
DOI: https://doi.org/10.1007/978-3-642-32512-0_45
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32511-3
Online ISBN: 978-3-642-32512-0
eBook Packages: Computer ScienceComputer Science (R0)