### 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(n^{1.5}+^{ε}) for any ε > 0.^{4} (ii) If C consists of two collections C_{1} and C_{2} where C_{1} 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 C_{2} is a collection of polygons with a total of n sides, then |R(C)| = O(n^{4/3}), 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(n^{2-1}/^{t}).

Original language | English (US) |
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Title of host publication | Algorithm Theory — SWAT 1998 - 6th Scandinavian Workshop on Algorithm Theory, Proceedings |

Publisher | Springer Verlag |

Pages | 322-334 |

Number of pages | 13 |

Volume | 1432 |

ISBN (Print) | 3540646825, 9783540646822 |

DOIs | |

State | Published - 1998 |

Event | 6th Scandinavian Workshop on Algorithm Theory, SWAT 1998 - Stockholm, Sweden Duration: Jul 8 1998 → Jul 10 1998 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1432 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 6th Scandinavian Workshop on Algorithm Theory, SWAT 1998 |
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Country | Sweden |

City | Stockholm |

Period | 7/8/98 → 7/10/98 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Algorithm Theory — SWAT 1998 - 6th Scandinavian Workshop on Algorithm Theory, Proceedings*(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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithm Theory — SWAT 1998 - 6th Scandinavian Workshop on Algorithm Theory, Proceedings.*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

}

TY - GEN

T1 - On the number of regular vertices of the union of Jordan Regions

AU - Aronov, Boris

AU - Efrat, Alon

AU - Halperin, Dan

AU - Sharir, Micha

PY - 1998

Y1 - 1998

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=84957696038&partnerID=8YFLogxK

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U2 - 10.1007/BFb0054379

DO - 10.1007/BFb0054379

M3 - Conference contribution

SN - 3540646825

SN - 9783540646822

VL - 1432

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 322

EP - 334

BT - Algorithm Theory — SWAT 1998 - 6th Scandinavian Workshop on Algorithm Theory, Proceedings

PB - Springer Verlag

ER -