Lower Bounds for Line Stabbing

We present an (n log n) xed order algebraic decision tree lower bound for determining the existence of a line stabber for a family of n line segments in the plane. We give the same lower bound for determining the existence of a line stabber for n translates of a circle in the plane. In proving this lower bound, we show that this problem is equivalent to determining if the width of a set of points is less than or equal to w. Through this transformation we can reexamine an old example by Hadwiger, Debrunner and Klee of a family of k + 1 translates where every k translates have a line transversal but the entire family has no line transversal. A line which intersects every member in a family of objects is known as a line stabber or line transversal. Recently, computer scientists have been very interested in algorithms for nding such line stabbers, In 8], Edelsbrunner et al. presented an O(n log n) algorithm for constructing a representation of the line stabbers of n line segments in the plane. This was generalized by Atallah and Bajaj to an O(nn(n) log n) algorithm for line stabbing n simple objects in the plane, where (n) is the inverse of Ackerman's function 1]. A simple object is an object which has an O(1) storage description and for which common tangents and intersections can be computed in O(1) time. Edelsbrunner, Guibas and Sharir showed how to construct a representation of the line stabbers of convex polygons with a total of n vertices in O(nn(n) log n) time 7]. Recently Hershberger 1