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 di erent stabbers for several shapes of stabbers. Speci cally, 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(n log n) (for strips, quadrants, and 3-sided rectangles), and O(n2 log n) (for rectangles).