Geometric hitting sets for disks

Theory and practice

Norbert Bus, Nabil H. Mustafa, Saurabh Ray

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    The geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a setD of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects inD. In 1994, Bronniman and Goodrich [6] made an important connection of this problem to the size of fundamental combinatorial structures called ϵ-nets, showing that small-sized ϵ-nets imply approximation algorithms with correspondingly small approximation ratios. Finally, recently Agarwal-Pan [5] showed that their scheme can be implemented in near-linear time for disks in the plane. This current state-of-the-art is lacking in three ways. First, the constants in current ϵ-net constructions are large, so the approximation factor ends up being more than 40. Second, the algorithm uses sophisticated geometric tools and data structures with large resulting constants. Third, these have resulted in a lack of available software for fast computation of small hitting-sets. In this paper, we make progress on all three of these barriers: i) we prove improved bounds on sizes of ϵ-nets, ii) design hitting-set algorithms without the use of these datastructures and finally, iii) present dnet, a public source-code module that incorporates both of these improvements to compute small-sized hitting sets and ϵ-nets efficiently in practice.

    Original languageEnglish (US)
    Title of host publicationAlgorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings
    PublisherSpringer-Verlag
    Pages903-914
    Number of pages12
    ISBN (Print)9783662483497
    DOIs
    StatePublished - Jan 1 2015
    Event23rd European Symposium on Algorithms, ESA 2015 - Patras, Greece
    Duration: Sep 14 2015Sep 16 2015

    Publication series

    NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
    Volume9294
    ISSN (Print)0302-9743
    ISSN (Electronic)1611-3349

    Other

    Other23rd European Symposium on Algorithms, ESA 2015
    CountryGreece
    CityPatras
    Period9/14/159/16/15

    Fingerprint

    Hitting Set
    Combinatorial optimization
    Approximation algorithms
    Data structures
    Data Structures
    Geometric Optimization
    Geometric object
    Approximation
    Combinatorial Optimization Problem
    Hits
    Linear Time
    Approximation Algorithms
    Imply
    Module
    Subset
    Software

    Keywords

    • Approximation Algorithms
    • Computational Geometry
    • Geometric Hitting Sets

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Computer Science(all)

    Cite this

    Bus, N., Mustafa, N. H., & Ray, S. (2015). Geometric hitting sets for disks: Theory and practice. In Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings (pp. 903-914). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9294). Springer-Verlag. https://doi.org/10.1007/978-3-662-48350-3_75

    Geometric hitting sets for disks : Theory and practice. / Bus, Norbert; Mustafa, Nabil H.; Ray, Saurabh.

    Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings. Springer-Verlag, 2015. p. 903-914 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9294).

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Bus, N, Mustafa, NH & Ray, S 2015, Geometric hitting sets for disks: Theory and practice. in Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9294, Springer-Verlag, pp. 903-914, 23rd European Symposium on Algorithms, ESA 2015, Patras, Greece, 9/14/15. https://doi.org/10.1007/978-3-662-48350-3_75
    Bus N, Mustafa NH, Ray S. Geometric hitting sets for disks: Theory and practice. In Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings. Springer-Verlag. 2015. p. 903-914. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-662-48350-3_75
    Bus, Norbert ; Mustafa, Nabil H. ; Ray, Saurabh. / Geometric hitting sets for disks : Theory and practice. Algorithms – ESA 2015 - 23rd Annual European Symposium, Proceedings. Springer-Verlag, 2015. pp. 903-914 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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