Location of weighted anti-ordered median straight lines with Euclidean distances

Given a set of nn weighted points in the plane, a set of non-negative (ordered) weights and a connected polygonal region SS, the weighted anti-ordered median straight-line location problem consists in finding a straight line intersecting SS and maximizing the sum of ordered weighted distances to the points. In this paper we show how to find such a straight line in O(n4)O(n4) time when the Euclidean distance is considered. As a consequence of the results given in the paper the weighted Euclidean anti-median straight-line problem can be solved in O(n2)O(n2) time.