---
_id: '4082'
abstract:
- lang: eng
text: Sweeping a collection of figures in the Euclidean plane with a straight line
is one of the novel algorithmic paradigms that have emerged in the field of computational
geometry. In this paper we demonstrate the advantages of sweeping with a topological
line that is not necessarily straight. We show how an arrangement of n lines in
the plane can be swept over in O(n2) time and O(n) space by a such a line. In
the process each element, i.e., vertex, edge, or region, is visited once in a
consistent ordering. Our technique makes use of novel data structures which exhibit
interesting amortized complexity behavior; the result is an algorithm that improves
upon all its predecessors either in the space or the time bounds, as well as being
eminently practical. Numerous applications of the technique to problems in computational
geometry are given—many through the use of duality transforms. Examples include
solving visibility problems, detecting degeneracies in configurations, computing
the extremal shadows of convex polytopes, and others. Even though our basic technique
solves a planar problem, its applications include several problems in higher dimensions.
acknowledgement: he authors wish to thank Raimund Seidel for suggesting the argument
that we used to prove Theorem 3.1, Harald Rosenberger who implemented the topological
sweep and compared it with a straight line sweep, the students who took the Stanford
1985 analysis of algorithms qualifying examination and suffered through a version
of this problem, and finally Lyle Ramshaw and Cynthia Hibbard for their detailed
reading and comments on the manuscript. The constructive criticism of an anonymous
referee is also appreciated.
article_processing_charge: No
article_type: original
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Leonidas
full_name: Guibas, Leonidas
last_name: Guibas
citation:
ama: Edelsbrunner H, Guibas L. Topologically sweeping an arrangement. Journal
of Computer and System Sciences. 1989;38(1):165-194. doi:10.1016/0022-0000(89)90038-X
apa: Edelsbrunner, H., & Guibas, L. (1989). Topologically sweeping an arrangement.
Journal of Computer and System Sciences. Elsevier. https://doi.org/10.1016/0022-0000(89)90038-X
chicago: Edelsbrunner, Herbert, and Leonidas Guibas. “Topologically Sweeping an
Arrangement.” Journal of Computer and System Sciences. Elsevier, 1989.
https://doi.org/10.1016/0022-0000(89)90038-X.
ieee: H. Edelsbrunner and L. Guibas, “Topologically sweeping an arrangement,” Journal
of Computer and System Sciences, vol. 38, no. 1. Elsevier, pp. 165–194, 1989.
ista: Edelsbrunner H, Guibas L. 1989. Topologically sweeping an arrangement. Journal
of Computer and System Sciences. 38(1), 165–194.
mla: Edelsbrunner, Herbert, and Leonidas Guibas. “Topologically Sweeping an Arrangement.”
Journal of Computer and System Sciences, vol. 38, no. 1, Elsevier, 1989,
pp. 165–94, doi:10.1016/0022-0000(89)90038-X.
short: H. Edelsbrunner, L. Guibas, Journal of Computer and System Sciences 38 (1989)
165–194.
date_created: 2018-12-11T12:06:50Z
date_published: 1989-02-01T00:00:00Z
date_updated: 2022-02-10T16:06:05Z
day: '01'
doi: 10.1016/0022-0000(89)90038-X
extern: '1'
intvolume: ' 38'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://www.sciencedirect.com/science/article/pii/002200008990038X?via%3Dihub
month: '02'
oa: 1
oa_version: Published Version
page: 165 - 194
publication: Journal of Computer and System Sciences
publication_identifier:
eissn:
- 1090-2724
issn:
- 0022-0000
publication_status: published
publisher: Elsevier
publist_id: '2039'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Topologically sweeping an arrangement
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 38
year: '1989'
...