Maximum - Weight Planar Boxes in O ( n 2 ) Time ( and Better ) I

Given a set P of n points in R, where each point p of P is associated with a weight w(p) (positive or negative), the Maximum-Weight Box problem is to find an axis-aligned box B maximizing ∑ p∈B∩P w(p). We describe algorithms for this problem in two dimensions that run in the worst case in O(n) time, and much less on more specific classes of instances. In particular, these results imply similar ones for the Maximum Bichromatic Discrepancy Box problem. These improve by a factor of Θ(lg n) on the previously known worst-case complexity for these problems, O(n lg n) [Cortés et al., J. Alg., 2009; Dobkin et al., J. Comput. Syst. Sci., 1996]. Although the O(n) result can be deduced from new results on Klee’s Measure problem [Chan, Proc. FOCS 2013], it is a more direct and simplified (non-trivial) solution. We exploit the connection with Klee’s Measure problem to further show that (1) the Maximum-Weight Box problem can be solved in O(n) time for any constant d ≥ 2; (2) if the weights are integers bounded by O(1) in absolute values, or weights are +1 and −∞ (as in the Maximum Bichromatic Discrepancy Box problem), the Maximum-Weight Box problem can be solved in O((n/ lg n)(lg lg n)) time; (3) it is unlikely that the Maximum-Weight Box problem can be solved in less than n time (ignoring logarithmic factors) with current knowledge about Klee’s Measure problem. IA previous version of this paper appeared in the Proceedings of the 25th Canadian Conference on Computational Geometry (CCCG’13) [2]. Email addresses: [email protected] (Jérémy Barbay), [email protected] (Timothy M. Chan), [email protected] (Gonzalo Navarro), [email protected] (Pablo Pérez-Lantero) 1Partially funded by Millennium Nucleus Information and Coordination in Networks ICM/FIC P10-024F, Mideplan, Chile. 2Partially supported by grant CONICYT, FONDECYT/Iniciación 11110069, Chile. Preprint submitted to Elsevier March 17, 2014