Abstract

Given a set P of n points in R-d, 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 Sigma(p is an element of B boolean AND P) w(P). We describe algorithms for this problem in two dimensions that run in the worst case in O(n(2)) 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 Theta(lg n) on the previously known worst-case complexity for these problems, O(n(2) lg n) (Cortes et al., 2009 [9]; Dobkin et al., 1996 [10]). Although the O(n(2)) 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(n(d)) time for any constant d >= 2; (2) if the weights are integers bounded by O(1) in absolute values, or weights are +1 and -infinity (as in the MAXIMUM BICHROMATIC DISCREPANCY Box problem), the MAXIMUM-WEIGHT Box problem can be solved in O((n(d)/lg(d) n)(lg lg n)(O(1))) time; (3) it is unlikely that the MAXIMUM-WEIGHT Box problem can be solved in less than n(d/2) time (ignoring logarithmic factors) with current knowledge about KLEE'S MEASURE problem. (C) 2014 Elsevier B.V. All rights reserved.