Abstract
Given a set R of r red points and a set B of b blue points in the plane, the static version of the Maximum Box Problem is to find an isothetic box H such that H∩R=∅ and the cardinality of H∩B is maximized. In this paper, we consider a kinetic version of the problem where the points in R∪B move along bounded degree algebraic trajectories. We design a compact and local quadratic-space kinetic data structure (KDS) for maintaining the optimal solution in O(rlog r+rlog b+b) time per each event. We also give an algorithm for solving the more general static problem where the maximum box can be arbitrarily oriented. This is an open problem in Aronov and Har-Peled (SIAM J. Comput. 38:899–921, 2008). We show that our approach can be used to solve this problem in O((r+b)2(rlog r+rlog b+b)) time. Finally we propose an efficient data structure to maintain an approximated solution of the kinetic Maximum Box Problem.
Similar content being viewed by others
References
Aronov B, Har-Peled S (2008) On approximating the depth and related problems. SIAM J Comput 38:899–921
Bonnet P, Gehrke J, Seshadri P (2001) Towards sensor database systems. In: Proc of the second international conf on mobile data management, Hong Kong. Lecture notes comp sci, vol 1987, pp 3–14
Chazelle B, Guibas LJ (1986) Fractional cascading: a data structuring technique. Algorithmica 1:133–162
Civilis A, Jensen CS, Pakalnis S (2005) Techniques for efficient road-network-based tracking of moving objects. IEEE Trans Knowl Data Eng 17(5):698–712
Cormen TH, Leiserson CE, Rivest RL (2001) Introduction to algorithms, 2nd edn. MIT Press, Cambridge
Cortés C, Díaz-Báñez JM, Pérez-Lantero P, Seara C, Urrutia J, Ventura I (2009) Bichromatic separability with two boxes: a general approach. J Algorithms, Cogn, Inf Log 64(2–3):79–88
Duda R, Hart P, Stork D (2001) Pattern classification. Wiley, New York
Guibas LJ (1998) Kinetic data structures: a state of the art report. In: Robotics: the algorithmic perspective. AK Peters, Wellesley, pp 191–209
Liu Y, Nediak M (2003) Planar case of the Maximum Box and related problems. In: Proc Canad conf comp geom, Halifax, Nova Scotia, August 11–13 (2003)
Segal M (2004) Planar Maximum Box Problem. J Math Modeling Algorithms 3:31–38
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by grant MEC MTM2009-08652.
Rights and permissions
About this article
Cite this article
Bereg, S., Díaz-Báñez, J.M., Pérez-Lantero, P. et al. The Maximum Box Problem for moving points in the plane. J Comb Optim 22, 517–530 (2011). https://doi.org/10.1007/s10878-010-9301-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-010-9301-2