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 contribution

1 Citation (Scopus)

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)|, 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 (US)
Title of host publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
PublisherSpringer Verlag
Pages322-334
Number of pages13
Volume1432
ISBN (Print)3540646825, 9783540646822
DOIs
StatePublished - 1998
Externally publishedYes
Event6th Scandinavian Workshop on Algorithm Theory, SWAT 1998 - Stockholm, Sweden
Duration: Jul 8 1998Jul 10 1998

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume1432
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other6th Scandinavian Workshop on Algorithm Theory, SWAT 1998
CountrySweden
CityStockholm
Period7/8/987/10/98

Fingerprint

Union
Intersect
Polygon
Arrangement
Intersection
Denote
Closed
Integer

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Aronov, B., Efrat, A., Halperin, D., & Sharir, M. (1998). On the number of regular vertices of the union of Jordan Regions. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1432, pp. 322-334). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1432). Springer Verlag. https://doi.org/10.1007/BFb0054379

On the number of regular vertices of the union of Jordan Regions. / Aronov, Boris; Efrat, Alon; Halperin, Dan; Sharir, Micha.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 1432 Springer Verlag, 1998. p. 322-334 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1432).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Aronov, B, Efrat, A, Halperin, D & Sharir, M 1998, On the number of regular vertices of the union of Jordan Regions. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 1432, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1432, Springer Verlag, pp. 322-334, 6th Scandinavian Workshop on Algorithm Theory, SWAT 1998, Stockholm, Sweden, 7/8/98. https://doi.org/10.1007/BFb0054379
Aronov B, Efrat A, Halperin D, Sharir M. On the number of regular vertices of the union of Jordan Regions. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 1432. Springer Verlag. 1998. p. 322-334. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/BFb0054379
Aronov, Boris ; Efrat, Alon ; Halperin, Dan ; Sharir, Micha. / On the number of regular vertices of the union of Jordan Regions. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 1432 Springer Verlag, 1998. pp. 322-334 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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