Maximum-Weight Planar Boxes in O(n2) Time (and Better)

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 consists in finding 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(log n) on the best worst-case complexity previously known 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 the Klee’s Measure problem [Chan, 2013], it is a more direct and simplified (nontrivial) solution, which further provides smaller running times on specific classes on instances.