The Maximum Box Problem for Moving Points in the Planeby S. Bereg, J. M. Díaz-Báñez, P. Pérez-Lantero, and I. Ventura.
Abstract: Given r red points and b blue points in the plane, the maximum box problem is to find an isothetic box containing maximum number of blue points and not containing red points. In Figure below, the optimal box contains 12 blue points.
In this paper, we consider a kinetic version of the problem where all the points 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.
pdf file submitted to Journal of Combinatorial Optimization.