Arrangements of segments that share endpoints: Single face results

Esther M. Arkin; Dan Halperin; Klara Kedem; Joseph S.B. Mitchell; Nir Naor

doi:10.1145/109648.109684

Arrangements of segments that share endpoints: Single face results

Esther M. Arkin, Dan Halperin, Klara Kedem, Joseph S.B. Mitchell, Nir Naor

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

Abstract

We provide new combinatorial bounds on the complexity of a face in an arrangement of segments in the plane. In particular, we show that the complexity of a single face in an arrangement of n line segments determined by h endpoints is ⊙(ha(h)). While the previous upper bound, O(nα(n)), is tight for segments with distinct endpoints, it is far from being optimal when n = Ω(h²). Our result shows that the fundamental combinatorial complexity of a face arises not as a result of the number of segments, but rather as a result of the number of endpoints. We generalize our bounds to the case of pseudosegments, and to the case of n chords in a polygon with h holes. Furthermore, our results lead to an improved algorithm for computing a single face in an arrangement of segments that share endpoints.

Original language	English
Title of host publication	Proceedings of the Annual Symposium on Computational Geometry
Publisher	Association for Computing Machinery
Pages	324-333
Number of pages	10
ISBN (Print)	0897914260
DOIs	https://doi.org/10.1145/109648.109684
State	Published - 1 Jun 1991
Externally published	Yes
Event	7th Annual Symposium on Computational Geometry, SCG 1991 - North Conway, United States Duration: 10 Jun 1991 → 12 Jun 1991

Publication series

Name	Proceedings of the Annual Symposium on Computational Geometry

Conference

Conference	7th Annual Symposium on Computational Geometry, SCG 1991
Country/Territory	United States
City	North Conway
Period	10/06/91 → 12/06/91

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Computational Mathematics

Access to Document

10.1145/109648.109684

Cite this

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title = "Arrangements of segments that share endpoints: Single face results",

abstract = "We provide new combinatorial bounds on the complexity of a face in an arrangement of segments in the plane. In particular, we show that the complexity of a single face in an arrangement of n line segments determined by h endpoints is ⊙(ha(h)). While the previous upper bound, O(nα(n)), is tight for segments with distinct endpoints, it is far from being optimal when n = Ω(h2). Our result shows that the fundamental combinatorial complexity of a face arises not as a result of the number of segments, but rather as a result of the number of endpoints. We generalize our bounds to the case of pseudosegments, and to the case of n chords in a polygon with h holes. Furthermore, our results lead to an improved algorithm for computing a single face in an arrangement of segments that share endpoints.",

author = "Arkin, {Esther M.} and Dan Halperin and Klara Kedem and Mitchell, {Joseph S.B.} and Nir Naor",

note = "Publisher Copyright: {\textcopyright} 1991 ACM.; 7th Annual Symposium on Computational Geometry, SCG 1991 ; Conference date: 10-06-1991 Through 12-06-1991",

year = "1991",

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doi = "10.1145/109648.109684",

language = "English",

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Arkin, EM, Halperin, D, Kedem, K, Mitchell, JSB & Naor, N 1991, Arrangements of segments that share endpoints: Single face results. in Proceedings of the Annual Symposium on Computational Geometry. Proceedings of the Annual Symposium on Computational Geometry, Association for Computing Machinery, pp. 324-333, 7th Annual Symposium on Computational Geometry, SCG 1991, North Conway, United States, 10/06/91. https://doi.org/10.1145/109648.109684

Arrangements of segments that share endpoints: Single face results. / Arkin, Esther M.; Halperin, Dan; Kedem, Klara et al.
Proceedings of the Annual Symposium on Computational Geometry. Association for Computing Machinery, 1991. p. 324-333 (Proceedings of the Annual Symposium on Computational Geometry).

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

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N2 - We provide new combinatorial bounds on the complexity of a face in an arrangement of segments in the plane. In particular, we show that the complexity of a single face in an arrangement of n line segments determined by h endpoints is ⊙(ha(h)). While the previous upper bound, O(nα(n)), is tight for segments with distinct endpoints, it is far from being optimal when n = Ω(h2). Our result shows that the fundamental combinatorial complexity of a face arises not as a result of the number of segments, but rather as a result of the number of endpoints. We generalize our bounds to the case of pseudosegments, and to the case of n chords in a polygon with h holes. Furthermore, our results lead to an improved algorithm for computing a single face in an arrangement of segments that share endpoints.

AB - We provide new combinatorial bounds on the complexity of a face in an arrangement of segments in the plane. In particular, we show that the complexity of a single face in an arrangement of n line segments determined by h endpoints is ⊙(ha(h)). While the previous upper bound, O(nα(n)), is tight for segments with distinct endpoints, it is far from being optimal when n = Ω(h2). Our result shows that the fundamental combinatorial complexity of a face arises not as a result of the number of segments, but rather as a result of the number of endpoints. We generalize our bounds to the case of pseudosegments, and to the case of n chords in a polygon with h holes. Furthermore, our results lead to an improved algorithm for computing a single face in an arrangement of segments that share endpoints.

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ER -

Arrangements of segments that share endpoints: Single face results

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