TY - JOUR

T1 - Separating and shattering long line segments

AU - Efrat, Alon

AU - Schwarzkopf, Otfried

N1 - Funding Information:
* Corresponding author. Email: alone@cs.tau.ac.il. ’ Work on this paper by the second author has been supported by the Netherlands’ Organization for Scientific Research (NWO), by Pohang University of Science and Technology Grant 96POO4, 1996, and partially by the nondirected research fund of the Korean Ministry of Education. * Email: otfried@postech.ac.kr.

PY - 1997/12/29

Y1 - 1997/12/29

N2 - 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 log n), improving (for this setting) the (deterministic) O(n2 log n) time algorithm of Freimer, Mitchell and Piatko.

AB - 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 log n), improving (for this setting) the (deterministic) O(n2 log n) time algorithm of Freimer, Mitchell and Piatko.

KW - BSP-trees

KW - Computational geometry

KW - Line separation

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U2 - 10.1016/s0020-0190(97)00188-9

DO - 10.1016/s0020-0190(97)00188-9

M3 - Article

AN - SCOPUS:0042634021

VL - 64

SP - 309

EP - 314

JO - Information Processing Letters

JF - Information Processing Letters

SN - 0020-0190

IS - 6

ER -