On the number of regular vertices of the union of Jordan Regions

Boris Aronov, Alon Efrat, Dan Halperin, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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)|, determined by the type of the sets of C. (i) If each set of C is convex, then [R(C)[ = O(n1.5+ε) for any ε > 0.4 (ii) If C consists of two collections C1 and C2 where C1 is a collection of n 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 C2 is a collection of polygons with a total of n sides, then |R(C)| = O(n4/3), and this bound is tight in the worst case. (iii) If no further assumptions are made on the sets of C, then we show that there is a positive integer t that depends only on s such that |R(C)| = O(n2-1/t).

Original languageEnglish
Title of host publicationAlgorithm Theory — SWAT 1998 - 6th Scandinavian Workshop on Algorithm Theory, Proceedings
EditorsStefan Arnborg, Lars Ivansson
PublisherSpringer Verlag
Number of pages13
ISBN (Print)3540646825, 9783540646822
StatePublished - 1998
Event6th Scandinavian Workshop on Algorithm Theory, SWAT 1998 - Stockholm, Sweden
Duration: 8 Jul 199810 Jul 1998

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference6th Scandinavian Workshop on Algorithm Theory, SWAT 1998


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