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A deterministic algorithm for partitioning arrangements of lines and its application

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P. K. AgarwalAuthors Info & Claims

SCG '89: Proceedings of the fifth annual symposium on Computational geometry

Pages 11 - 22

https://doi.org/10.1145/73833.73835

Published: 05 June 1989 Publication History

Abstract

In this paper we consider the following problem: Given a set ℒ of n lines in the plane, partition the plane into Ο(r²) triangles so that no triangle intersects more than Ο(n/r) lines of ℒ. We present a deterministic algorithm for this problem with Ο(nr log n log^ω r) running time, where ω is a constant < 3.3. Our algorithm is faster than Matousk's recent algorithm [Ma] for large values of r. In the second part of the paper, we apply this algorithm to several problems involving lines or segments in the plane, and obtain deterministic algorithms which are faster than any previously known algorithms. For example we give an Ο(n^2/3m^2/3 log n log^ω/3 m/√n + (m + n) log n) algorithm to compute all incidences between m points and n lines. Other problems include computing many faces in an arrangement of lines or segments, counting segment intersections, red-blue intersection detection, simplex range queries and computing stabbing trees with low stabbing number.

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Cited By

Matoušek J(2011)Spanning trees with low crossing numberRAIRO - Theoretical Informatics and Applications10.1051/ita/199125020103125:2(103-123)Online publication date: 8-Jan-2011
https://doi.org/10.1051/ita/1991250201031
Segal MZeitlin E(2008)Computing closest and farthest points for a query segmentTheoretical Computer Science10.1016/j.tcs.2007.11.015393:1-3(294-300)Online publication date: 1-Mar-2008
https://dl.acm.org/doi/10.1016/j.tcs.2007.11.015
Schipper HOvermars M(2005)Dynamic partition treesSWAT 9010.1007/3-540-52846-6_108(404-417)Online publication date: 8-Jun-2005
https://doi.org/10.1007/3-540-52846-6_108
Show More Cited By

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A deterministic algorithm for partitioning arrangements of lines and its application

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Partitioning arrangements of lines II: Applications

In this paper we present efficient deterministic algorithms for various problems involving lines or segments in the plane, using the partitioning algorithm described in a companion paper [A3]. These applications include: (i) anO(m2/3n2/3 log2/3n log /3 (...
Partitioning arrangements of lines I: An efficient deterministic algorithm

In this paper we consider the following problem: Given a set ofn lines in the plane, partition the plane intoO(r2) triangles so that no triangle meets more thanO(n/r) lines of . We present a deterministic algorithm for this problem withO(nr logn/r) ...
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cover image ACM Conferences

SCG '89: Proceedings of the fifth annual symposium on Computational geometry

June 1989

401 pages

ISBN:0897913183

DOI:10.1145/73833

Chairman:
Kurt Mehlhorn

Copyright © 1989 ACM.

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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Published: 05 June 1989

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Cited By

Matoušek J(2011)Spanning trees with low crossing numberRAIRO - Theoretical Informatics and Applications10.1051/ita/199125020103125:2(103-123)Online publication date: 8-Jan-2011
https://doi.org/10.1051/ita/1991250201031
Segal MZeitlin E(2008)Computing closest and farthest points for a query segmentTheoretical Computer Science10.1016/j.tcs.2007.11.015393:1-3(294-300)Online publication date: 1-Mar-2008
https://dl.acm.org/doi/10.1016/j.tcs.2007.11.015
Schipper HOvermars M(2005)Dynamic partition treesSWAT 9010.1007/3-540-52846-6_108(404-417)Online publication date: 8-Jun-2005
https://doi.org/10.1007/3-540-52846-6_108
Mitchell JMount DSuri SMehlhorn K(1994)Query-sensitive ray shootingProceedings of the tenth annual symposium on Computational geometry10.1145/177424.178094(359-368)Online publication date: 10-Jun-1994
https://dl.acm.org/doi/10.1145/177424.178094
Pellegrini MShor P(1992)Finding stabbing lines in 3-spaceDiscrete & Computational Geometry10.1007/BF022930438:2(191-208)Online publication date: 1-Dec-1992
https://dl.acm.org/doi/10.1007/BF02293043
Cheng SJanardan RAggarwal A(1991)Space-efficient ray-shooting and intersection searchingProceedings of the second annual ACM-SIAM symposium on Discrete algorithms10.5555/127787.127797(7-16)Online publication date: 1-Mar-1991
https://dl.acm.org/doi/10.5555/127787.127797
Pellegrini MShor PAggarwal A(1991)Finding stabbing lines in 3-dimensional spaceProceedings of the second annual ACM-SIAM symposium on Discrete algorithms10.5555/127787.126018(24-31)Online publication date: 1-Mar-1991
https://dl.acm.org/doi/10.5555/127787.126018
Goodrich MLeighton T(1991)Constructing arrangements optimally in parallel (preliminary version)Proceedings of the third annual ACM symposium on Parallel algorithms and architectures10.1145/113379.113395(169-179)Online publication date: 1-Jun-1991
https://dl.acm.org/doi/10.1145/113379.113395
Agarwal Pvan Kreveld MOvermars MDrysdale R(1991)Intersection queries for curved objects (extended abstract)Proceedings of the seventh annual symposium on Computational geometry10.1145/109648.109653(41-50)Online publication date: 1-Jun-1991
https://dl.acm.org/doi/10.1145/109648.109653
Agarwal PSharir M(1991)Applications of a new space partitioning techniqueAlgorithms and Data Structures10.1007/BFb0028277(379-391)Online publication date: 1991
https://doi.org/10.1007/BFb0028277
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