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 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(n 2 log n) time algorithm of Freimer, Mitchell and Piatko.
Work on this paper by the second author has been supported by the Netherlands' Organization for Scientific Research (NWO), and partially by Pohang University of Science and Technology Grant P96005, 1996.
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© 1996 Springer-Verlag Berlin Heidelberg
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Efrat, A., Schwarzkopf, O. (1996). Separating and shattering long line segments. In: Asano, T., Igarashi, Y., Nagamochi, H., Miyano, S., Suri, S. (eds) Algorithms and Computation. ISAAC 1996. Lecture Notes in Computer Science, vol 1178. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0009479
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DOI: https://doi.org/10.1007/BFb0009479
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