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Diameter, width, closest line pair, and parametric searching

Authors:

Bernard Chazelle,

Herbert Edelsbrunner,

Leonidas Guibas,

Micha SharirAuthors Info & Claims

SCG '92: Proceedings of the eighth annual symposium on Computational geometry

Pages 120 - 129

https://doi.org/10.1145/142675.142702

Published: 01 July 1992 Publication History

Abstract

We apply Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improve solutions for them. We obtain, for any fixed ε > 0, an O(n^1+ε) algorithm for computing the diameter of a point set in 3-space, an O(n^8/5+ε) algorithm for computing the closest pair in a set of n lines in space. All these algorithms are deterministic. We also look at the problem of computing the k-th smallest slope formed by the lines joining n points in the plane. In 1989 Cole, Salowe, Steiger, and Szemere´di gave an optimal but very complicated O(n log n) solution based on Megiddo's technique. We follow a different route and give a very simple O(n log² n) solution which bypasses parametric searching altogether.

References

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Cited By

Hassanat A(2018)Greedy Algorithms for Approximating the Diameter of Machine Learning Datasets in Multidimensional Euclidean Space: Experimental ResultsADCAIJ: Advances in Distributed Computing and Artificial Intelligence Journal10.14201/ADCAIJ20187315307:3(15-30)Online publication date: 13-Sep-2018
https://doi.org/10.14201/ADCAIJ2018731530
Mckenney MFrye RBenchly ZMaughan L(2017)Operations to support temporal coverage aggregates over moving regionsGeoinformatica10.1007/s10707-016-0257-y21:2(351-364)Online publication date: 1-Apr-2017
https://dl.acm.org/doi/10.1007/s10707-016-0257-y
McKenney MFrye RBenchly ZMaughan LNguyen P(2014)Temporal coverage aggregates over moving region streamsProceedings of the 5th ACM SIGSPATIAL International Workshop on GeoStreaming10.1145/2676552.2676555(21-24)Online publication date: 4-Nov-2014
https://dl.acm.org/doi/10.1145/2676552.2676555
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Index Terms

Diameter, width, closest line pair, and parametric searching
1. Mathematics of computing
  1. Mathematical analysis
    1. Mathematical optimization
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
  2. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SCG '92: Proceedings of the eighth annual symposium on Computational geometry

July 1992

384 pages

ISBN:0897915178

DOI:10.1145/142675

Chairman:
David Avis

Copyright © 1992 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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Publication History

Published: 01 July 1992

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SOCG92

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SOCG92: 8th Annual Symposium on Computational Geometry 1992

June 10 - 12, 1992

Berlin, Germany

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Overall Acceptance Rate 625 of 1,685 submissions, 37%

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Cited By

Hassanat A(2018)Greedy Algorithms for Approximating the Diameter of Machine Learning Datasets in Multidimensional Euclidean Space: Experimental ResultsADCAIJ: Advances in Distributed Computing and Artificial Intelligence Journal10.14201/ADCAIJ20187315307:3(15-30)Online publication date: 13-Sep-2018
https://doi.org/10.14201/ADCAIJ2018731530
Mckenney MFrye RBenchly ZMaughan L(2017)Operations to support temporal coverage aggregates over moving regionsGeoinformatica10.1007/s10707-016-0257-y21:2(351-364)Online publication date: 1-Apr-2017
https://dl.acm.org/doi/10.1007/s10707-016-0257-y
McKenney MFrye RBenchly ZMaughan LNguyen P(2014)Temporal coverage aggregates over moving region streamsProceedings of the 5th ACM SIGSPATIAL International Workshop on GeoStreaming10.1145/2676552.2676555(21-24)Online publication date: 4-Nov-2014
https://dl.acm.org/doi/10.1145/2676552.2676555
Gupta PJanardan RSmid M(2006)Fast algorithms for collision and proximity problems involving moving geometric objectsAlgorithms — ESA '9410.1007/BFb0049415(278-289)Online publication date: 23-Feb-2006
https://doi.org/10.1007/BFb0049415
HURTADO FSEARA CSETHIA S(2005)RED-BLUE SEPARABILITY PROBLEMS IN 3DInternational Journal of Computational Geometry & Applications10.1142/S021819590500164615:02(167-192)Online publication date: Apr-2005
https://doi.org/10.1142/S0218195905001646
Fernández-Baca DSlutzki GEppstein D(2005)Using sparsification for parametric minimum spanning tree problemsAlgorithm Theory — SWAT'9610.1007/3-540-61422-2_128(149-160)Online publication date: 7-Jun-2005
https://doi.org/10.1007/3-540-61422-2_128
Fernández-Baca DSlutzki G(2005)Linear-time algorithms for parametric minimum spanning tree problems on planar graphsLATIN '95: Theoretical Informatics10.1007/3-540-59175-3_94(257-271)Online publication date: 6-Jun-2005
https://doi.org/10.1007/3-540-59175-3_94
Fernández-Baca DSlutzki G(2005)Optimal parametric search on graphs of bounded tree-widthAlgorithm Theory — SWAT '9410.1007/3-540-58218-5_14(155-166)Online publication date: 30-May-2005
https://doi.org/10.1007/3-540-58218-5_14
Kim YLin MManocha D(2004)Incremental Penetration Depth Estimation between Convex Polytopes Using Dual-Space ExpansionIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2004.126076710:2(152-163)Online publication date: 1-Mar-2004
https://dl.acm.org/doi/10.1109/TVCG.2004.1260767
Schieber B(1998)Computing a Minimum Weightk-Link Path in Graphs with the Concave Monge PropertyJournal of Algorithms10.1006/jagm.1998.095529:2(204-222)Online publication date: 1-Nov-1998
https://dl.acm.org/doi/10.1006/jagm.1998.0955
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