Improved Bounds on the Spanning Ratio of the Theta-5-Graph

We show an upper bound of sin3π10sin2π5-sin3π10<5.70 $$ \frac{\sin \left( \frac{3\pi }{10}\right) }{\sin \left( \frac{2\pi }{5}\right) -\sin \left( \frac{3\pi }{10}\right) } <5.70$$ on the spanning ratio of Θ5 $$\varTheta _5$$ -graphs, improving on the previous best known upper bound of 9.96 $...

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Bibliographic Details
Published inAlgorithms and Data Structures Vol. 12808; pp. 215 - 228
Main Authors Bose, Prosenjit, Hill, Darryl, Ooms, Aurélien
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 2021
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Geometric spanners
Spanning ratio
Stretch factor
Theta graphs
Online AccessGet full text
ISBN3030835073
9783030835071
ISSN0302-9743
1611-3349
DOI10.1007/978-3-030-83508-8_16

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Summary:We show an upper bound of sin3π10sin2π5-sin3π10<5.70 $$ \frac{\sin \left( \frac{3\pi }{10}\right) }{\sin \left( \frac{2\pi }{5}\right) -\sin \left( \frac{3\pi }{10}\right) } <5.70$$ on the spanning ratio of Θ5 $$\varTheta _5$$ -graphs, improving on the previous best known upper bound of 9.96 $$9.96$$ [Bose, Morin, van Renssen, and Verdonschot. The Theta-5-graph is a spanner. Computational Geometry, 2015.]
Bibliography:Research supported in part by NSERC, VILLUM Foundation grant 16582, and FRIA Grant 5203818F (FNRS).
Original Abstract: We show an upper bound of sin3π10sin2π5-sin3π10<5.70\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\sin \left( \frac{3\pi }{10}\right) }{\sin \left( \frac{2\pi }{5}\right) -\sin \left( \frac{3\pi }{10}\right) } <5.70$$\end{document} on the spanning ratio of Θ5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varTheta _5$$\end{document}-graphs, improving on the previous best known upper bound of 9.96\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$9.96$$\end{document} [Bose, Morin, van Renssen, and Verdonschot. The Theta-5-graph is a spanner. Computational Geometry, 2015.]
ISBN:3030835073
9783030835071
ISSN:0302-9743
1611-3349
DOI:10.1007/978-3-030-83508-8_16