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An expander-based approach to geometric optimization

Authors:

Matthew J. Katz,

Micha SharirAuthors Info & Claims

SCG '93: Proceedings of the ninth annual symposium on Computational geometry

Pages 198 - 207

https://doi.org/10.1145/160985.161137

Published: 01 July 1993 Publication History

Abstract

We present a new approach to problems in geometric optimization that are traditionally solved using the parametric searching technique of Megiddo. Our new approach is based on expander graphs and is conceptually much simpler and has more explicit geometric flavor. It does not require parallelization or randomization, and it exploits recent range-searching techniques of Matousˇek and others. We exemplify the technique on three problems, the slope selection problem, the planar distance selection problem, and the planar two-center problem. For the first problem we develop an O(n log³n)) solution, which, although suboptimal, is very simple. The second and third problems are more typical examples of our approach. Our solutions have, respectively, running time O(n^4/3 log^3+δ n), for any δ > 0, and O(n² log³ n), comparable with the respective solutions of [2, 5].

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A. Lubotzky, R. Phillips and P. Sarnak, Explicit expanders and the Ramanujan conjectures, Proc. ~!8th A CM Syrup. on Theory o/Computing, 1986, 240-(.!46. See also: Ramanujan graphs, Combinatorica 8 (1988), 261-277.

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Cited By

Choi JLee JAhn H(2023)Efficient k-Center Algorithms for Planar Points in Convex PositionAlgorithms and Data Structures10.1007/978-3-031-38906-1_18(262-274)Online publication date: 28-Jul-2023
https://doi.org/10.1007/978-3-031-38906-1_18
Tan XWu G(2010)New algorithms for barrier coverage with mobile sensorsProceedings of the 4th international conference on Frontiers in algorithmics10.5555/1881195.1881226(327-338)Online publication date: 11-Aug-2010
https://dl.acm.org/doi/10.5555/1881195.1881226
Tan XWu G(2010)New Algorithms for Barrier Coverage with Mobile SensorsFrontiers in Algorithmics10.1007/978-3-642-14553-7_31(327-338)Online publication date: 2010
https://doi.org/10.1007/978-3-642-14553-7_31
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Index Terms

An expander-based approach to geometric optimization
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
  2. Mathematical analysis
    1. Mathematical optimization
2. Theory of computation
  1. Design and analysis of algorithms
    1. Data structures design and analysis
      1. Sorting and searching
    2. Mathematical optimization
  2. Randomness, geometry and discrete structures
    1. Computational geometry

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cover image ACM Conferences

SCG '93: Proceedings of the ninth annual symposium on Computational geometry

July 1993

406 pages

ISBN:0897915828

DOI:10.1145/160985

Chairman:
Chee Yap

Copyright © 1993 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 1993

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9SCG93: Ninth Symposium on Computational Geometry

May 18 - 21, 1993

California, San Diego, USA

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

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Cited By

Choi JLee JAhn H(2023)Efficient k-Center Algorithms for Planar Points in Convex PositionAlgorithms and Data Structures10.1007/978-3-031-38906-1_18(262-274)Online publication date: 28-Jul-2023
https://doi.org/10.1007/978-3-031-38906-1_18
Tan XWu G(2010)New algorithms for barrier coverage with mobile sensorsProceedings of the 4th international conference on Frontiers in algorithmics10.5555/1881195.1881226(327-338)Online publication date: 11-Aug-2010
https://dl.acm.org/doi/10.5555/1881195.1881226
Tan XWu G(2010)New Algorithms for Barrier Coverage with Mobile SensorsFrontiers in Algorithmics10.1007/978-3-642-14553-7_31(327-338)Online publication date: 2010
https://doi.org/10.1007/978-3-642-14553-7_31
Agarwal PSharir M(2005)Algorithmic techniques for geometric optimizationComputer Science Today10.1007/BFb0015247(234-253)Online publication date: 9-Jun-2005
https://doi.org/10.1007/BFb0015247
Devillers OKatz M(2005)Optimal line bipartitions of point setsAlgorithms and Computation10.1007/BFb0009480(45-54)Online publication date: 11-Oct-2005
https://doi.org/10.1007/BFb0009480
Sharir M(2005)Arrangements in higher dimensions: Voronoi diagrams, motion planning, and other applicationsAlgorithms and Data Structures10.1007/3-540-60220-8_55(109-121)Online publication date: 1-Jun-2005
https://doi.org/10.1007/3-540-60220-8_55
Glozman AKedem KShpitalnik G(2005)On some geometric selection and optimization problems via sorted matricesAlgorithms and Data Structures10.1007/3-540-60220-8_48(26-37)Online publication date: 1-Jun-2005
https://doi.org/10.1007/3-540-60220-8_48
DEVILLERS OKATZ M(1999)OPTIMAL LINE BIPARTITIONS OF POINT SETSInternational Journal of Computational Geometry & Applications10.1142/S021819599900004209:01(39-51)Online publication date: Feb-1999
https://doi.org/10.1142/S0218195999000042
Chan TJanardan R(1998)On enumerating and selecting distancesProceedings of the fourteenth annual symposium on Computational geometry10.1145/276884.276916(279-286)Online publication date: 7-Jun-1998
https://dl.acm.org/doi/10.1145/276884.276916
Eppstein DSaks M(1997)Faster construction of planar two-centersProceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms10.5555/314161.314198(131-138)Online publication date: 5-Jan-1997
https://dl.acm.org/doi/10.5555/314161.314198
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