Separating and Shattering Long Line Segments ?

A line l is called a separator for a set S of objects in the plane if l avoids all the objects and partitions S into two non-empty subsets, lying on both sides of l. A set L of lines is said to shatter S if each line of L is a separator for S, and every two objects of S are separated by at least one line of L. We give a simple randomized algorithm to construct the set of all separators for a given set S of n line segments in expected time O(n log n), provided the ratio between the diameter of S and the length of the shortest line segment is bounded by a constant. We also give a randomized algorithm to determine a set of lines shattering S, whose expected running time is O(n logn), improving (for this setting) the (deterministic) O(n 2 log n) time algorithm of Freimer, Mitchell and Piatko.