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An optimal algorithm for intersecting line segments in the plane

Authors:

Bernard Chazelle,

Herbert EdelsbrunnerAuthors Info & Claims

Journal of the ACM (JACM), Volume 39, Issue 1

Pages 1 - 54

https://doi.org/10.1145/147508.147511

Published: 02 January 1992 Publication History

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Abstract

The main contribution of this work is an O(n log n + k)-time algorithm for computing all k intersections among n line segments in the plane. This time complexity is easily shown to be optimal. Within the same asymptotic cost, our algorithm can also construct the subdivision of the plane defined by the segments and compute which segment (if any) lies right above (or below) each intersection and each endpoint. The algorithm has been implemented and performs very well. The storage requirement is on the order of n + k in the worst case, but it is considerably lower in practice. To analyze the complexity of the algorithm, an amortization argument based on a new combinatorial theorem on line arrangements is used.

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Cited By

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Frye RMcKenney M(2024)Per Segment Plane Sweep Line Segment Intersection on the GPUACM Transactions on Spatial Algorithms and Systems10.1145/3701989Online publication date: 28-Oct-2024
https://dl.acm.org/doi/10.1145/3701989
Jiang JZou LZhao WHe ZChen TYu BDe V(2024)PDRC: Package Design Rule Checking via GPU-Accelerated Geometric Intersection Algorithms for Non-Manhattan GeometryProceedings of the 61st ACM/IEEE Design Automation Conference10.1145/3649329.3657367(1-6)Online publication date: 23-Jun-2024
https://dl.acm.org/doi/10.1145/3649329.3657367
Davoodi MSafari A(2024)Path Planning in a Weighted Planar Subdivision Under the Manhattan MetricGraphs and Combinatorics10.1007/s00373-023-02744-740:1Online publication date: 19-Jan-2024
https://dl.acm.org/doi/10.1007/s00373-023-02744-7
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Index Terms

An optimal algorithm for intersecting line segments in the plane
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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Reviews

Reviewer: John B. Slater

A need to calculate efficiently the intersection points of sets of line segments arises naturally from the manipulation of display and related entities through, for instance, windowing, clipping, and hidden surface removal. This research paper introduces a new algorithm leading to an essentially optimal single-processor time complexity, although some space considerations are left undecided. The algorithm is a complicated one, based on a basic sweep technique using end points only in the schedule, analyzing the intersections of segments in front of the sweep line using information derived from its previous history. A good survey of all the relevant concepts in the area is given, providing a good background to recent work in this area. This background is backed up by well-chosen references. The specific geometrical knowledge required to follow the algorithm is essentially elementary, but a thorough understanding of complexity and a good working knowledge of a number of areas of computing algorithms and theory is required, as is geometrical insight. The algorithm, implemented in C, shows considerable ingenuity to reach the eventual complexity of O n log n+k time and O n+k space to compute the k intersections of n line segments. The space complexity is not demonstrably optimal, and possible improvement is discussed. Nevertheless, the result is excellent and well presented.

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Published In

Journal of the ACM Volume 39, Issue 1

Jan. 1992

251 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/147508

Editor:
F. Thomson Leighton
Massachusetts Institute of Technology, Cambridge

Issue’s Table of Contents

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 02 January 1992

Published in JACM Volume 39, Issue 1

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https://dl.acm.org/doi/10.1145/3701989
Jiang JZou LZhao WHe ZChen TYu BDe V(2024)PDRC: Package Design Rule Checking via GPU-Accelerated Geometric Intersection Algorithms for Non-Manhattan GeometryProceedings of the 61st ACM/IEEE Design Automation Conference10.1145/3649329.3657367(1-6)Online publication date: 23-Jun-2024
https://dl.acm.org/doi/10.1145/3649329.3657367
Davoodi MSafari A(2024)Path Planning in a Weighted Planar Subdivision Under the Manhattan MetricGraphs and Combinatorics10.1007/s00373-023-02744-740:1Online publication date: 19-Jan-2024
https://dl.acm.org/doi/10.1007/s00373-023-02744-7
Schwarz NStorandt S(2024)Efficient Computation of Crossing Components and Shortcut HullsCombinatorial Algorithms10.1007/978-3-031-63021-7_39(509-522)Online publication date: 1-Jul-2024
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