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 sub sets 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 be tween 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 log n time algorithm of Freimer Mitchell and Piatko