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Workshop on Algorithms and Data Structures

WADS 2021: Algorithms and Data Structures pp 1–14Cite as

On the Spanning and Routing Ratios of the Directed \(\varTheta _6\)-Graph

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12808))

Abstract

The family of \(\varTheta _k\)-graphs is an important class of sparse geometric spanners with a small spanning ratio. Although they are a well-studied class of geometric graphs, no bound is known on the spanning and routing ratios of the directed \(\varTheta _6\)-graph. We show that the directed \(\varTheta _6\)-graph of a point set P, denoted \(\overrightarrow{\varTheta }_6(P)\), is a 7-spanner and there exist point sets where the spanning ratio is at least \(4-\varepsilon \), for any \(\varepsilon >0\). It is known that the standard greedy \(\varTheta \)-routing algorithm may have an unbounded routing ratio on \(\overrightarrow{\varTheta }_6(P)\). We design a simple, online, local, memoryless routing algorithm on \(\overrightarrow{\varTheta }_6(P)\) whose routing ratio is at most 14 and show that no algorithm can have a routing ratio better than \(6-\varepsilon \).

Research supported in part by NSERC.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Massachusetts Lowell, Lowell, USA

    Hugo A. Akitaya

  2. School of Computer Science, University of Windsor, Windsor, Canada

    Ahmad Biniaz

  3. School of Computer Science, Carleton University, Ottawa, Canada

    Prosenjit Bose

Authors
  1. Hugo A. Akitaya
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  2. Ahmad Biniaz
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  3. Prosenjit Bose
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Correspondence to Prosenjit Bose .

Editor information

Editors and Affiliations

  1. University of Waterloo, Waterloo, ON, Canada

    Anna Lubiw

  2. University of Alberta, Edmonton, AB, Canada

    Mohammad Salavatipour

  3. Dalhousie University, Halifax, Canada

    Meng He

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Akitaya, H.A., Biniaz, A., Bose, P. (2021). On the Spanning and Routing Ratios of the Directed \(\varTheta _6\)-Graph. In: Lubiw, A., Salavatipour, M., He, M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science(), vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_1

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  • DOI: https://doi.org/10.1007/978-3-030-83508-8_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-83507-1

  • Online ISBN: 978-3-030-83508-8

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