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Lower Bounds on the Dilation of Plane Spanners

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Algorithms and Discrete Applied Mathematics (CALDAM 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9602))

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Abstract

(I) We exhibit a set of 23 points in the plane that has dilation at least 1.4308, improving the previously best lower bound of 1.4161 for the worst-case dilation of plane spanners.

(II) For every \(n\ge 13\), there exists an n-element point set S such that the degree 3 dilation of S denoted by \(\delta _0(S,3) \text { equals } 1+\sqrt{3}=2.7321\ldots \) in the domain of plane geometric spanners. In the same domain, we show that for every \(n\ge 6\), there exists a an n-element point set S such that the degree 4 dilation of S denoted by \(\delta _0(S,4) \text { equals } 1 + \sqrt{(5-\sqrt{5})/2}=2.1755\ldots \) The previous best lower bound of 1.4161 holds for any degree.

(III) For every \(n\ge 6 \), there exists an n-element point set S such that the stretch factor of the greedy triangulation of S is at least 2.0268.

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Notes

  1. 1.

    Due to page limitations, the proofs of Lemmas 2, 3, 4 are deferred to the full version [21].

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Authors and Affiliations

  1. Department of Computer Science, University of Wisconsin-Milwaukee, Milwaukee, WI, 53201-0784, USA

    Adrian Dumitrescu & Anirban Ghosh

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  1. Adrian Dumitrescu
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  2. Anirban Ghosh
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Correspondence to Adrian Dumitrescu .

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Editors and Affiliations

  1. Indian Institute of Science, Bangalore, India

    Sathish Govindarajan

  2. Carleton University, Ottawa, ON, Canada

    Anil Maheshwari

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Dumitrescu, A., Ghosh, A. (2016). Lower Bounds on the Dilation of Plane Spanners. In: Govindarajan, S., Maheshwari, A. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2016. Lecture Notes in Computer Science(), vol 9602. Springer, Cham. https://doi.org/10.1007/978-3-319-29221-2_12

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  • DOI: https://doi.org/10.1007/978-3-319-29221-2_12

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