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 algorithm to construct the set of all separators for a given set S of n line segments in 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 an O(n log n)-time algorithm to determine a set of lines shattering S, improving (for this setting) the O(n2 log n) time algorithm of Freimer, Mitchell and Piatko.
KeywordsComputational Geometry BSP-trees line-seperation
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