The Maximum Box Problem for moving points in the plane
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- Bereg, S., Díaz-Báñez, J.M., Pérez-Lantero, P. et al. J Comb Optim (2011) 22: 517. doi:10.1007/s10878-010-9301-2
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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.