Maximum-weight planar boxes in O(n2)O(n2) time (and better)

• n points in the plane, each with a positive or negative weight, are considered.
• A maximum box is an axis-aligned rectangle covering the maximum weight sum.
• We find a maximum box in O(n2)O(n2) time improving upon previous O(n2logn) algorithms.
• We provide adaptive algorithms improving the complexity on specific instances.
• We show extensions in higher dimensions and the relation with Klee's Measure problem.

Given a set P of n   points in RdRd, where each point p of P   is associated with a weight w(p)w(p) (positive or negative), the Maximum-Weight Box problem is to find an axis-aligned box B   maximizing ∑p∈B∩Pw(p)∑p∈B∩Pw(p). We describe algorithms for this problem in two dimensions that run in the worst case in O(n2)O(n2) 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(lgn) on the previously known worst-case complexity for these problems, O(n2lgn) (Cortés et al., 2009 [9]; Dobkin et al., 1996 [10]). Although the O(n2)O(n2) result can be deduced from new results on Klee's Measure problem (Chan, 2013 [7]), 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(nd)O(nd) time for any constant d≥2d≥2; (2) if the weights are integers bounded by O(1)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((nd/lgdn)(lglgn)O(1)) time; (3) it is unlikely that the Maximum-Weight Box problem can be solved in less than nd/2nd/2 time (ignoring logarithmic factors) with current knowledge about Klee's Measure problem.