Stabbing segments with rectilinear objects

Given a set S of n line segments in the plane, we say that a region R⊆R² is a stabber for S if R contains exactly one endpoint of each segment of S. In this paper we provide optimal or near-optimal algorithms for reporting all combinatorially different stabbers for several shapes of stabbers. Specifically, we consider the case in which the stabber can be described as the intersection of axis-parallel halfplanes (thus the stabbers are halfplanes, strips, quadrants, 3-sided rectangles, or rectangles). The running times are O(n) (for the halfplane case), O(nlog n) (for strips, quadrants, and 3-sided rectangles), and O(n²log n) (for rectangles).