Separating and shattering long line segments

Abstract

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 ail 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 log n), improving (for this setting) the (deterministic) O(n(2) log n) time algorithm of Freimer, Mitchell and Piatko. (C) 1997 Published by Elsevier Science B.V.