On the number of regular vertices of the union of jordan regions |
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Authors: | B Aronov A Efrat D Halperin M Sharir |
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Institution: | (1) Department of Computer and Information Science, Polytechnic University, Brooklyn, NY 11201-3840, USA aronov@ziggy.poly.edu , US;(2) Computer Science Department, Stanford University, Stanford, CA 94305, USA alon@cs.stanford.edu , US;(3) School of Mathematical Sciences, Tel Aviv University, Tel-Aviv 69978, Israel \{halperin,sharir\}@math.tau.ac.il , IL;(4) Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA, US |
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Abstract: | Let \C be a collection of n Jordan regions in the plane in general position, such that each pair of their boundaries intersect in at most s points, where s is a constant. If the boundaries of two sets in \C cross exactly twice, then their intersection points are called regular vertices of the arrangement \A(\C) . Let R(\C) denote the set of regular vertices on the boundary of the union of \C . We present several bounds on |R(\C)| , depending on the type of the sets of \C . (i) If each set of \C is convex, then |R(\C)|=O(n
1.5+\eps
) for any \eps>0 . (ii) If no further assumptions are made on the sets of \C , then we show that there is a positive integer r that depends only on s such that |R(\C)|=O(n
2-1/r
) . (iii) If \C consists of two collections \C
1
and \C
2
where \C
1
is a collection of m convex pseudo-disks in the plane (closed Jordan regions with the property that the boundaries of any two of them intersect at most twice), and
\C
2
is a collection of polygons with a total of n sides, then |R(\C)|=O(m
2/3
n
2/3
+m +n) , and this bound is tight in the worst case.
Received December 4, 1998, and in revised form June 3, 2000. Online publication Feburary 1, 2001. |
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Keywords: | |
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