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Workshop on Algorithms and Data Structures

WADS 2019: Algorithms and Data Structures pp 466–480Cite as

A Linear-Time Algorithm for Radius-Optimally Augmenting Paths in a Metric Space

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11646))

Abstract

Let P be a path graph of n vertices embedded in a metric space. We consider the problem of adding a new edge to P to minimize the radius of the resulting graph. Previously, a similar problem for minimizing the diameter of the graph was solved in \(O(n\log n)\) time. To the best of our knowledge, the problem of minimizing the radius has not been studied before. In this paper, we present an O(n) time algorithm for the problem, which is optimal.

A full version of this paper is available at https://arxiv.org/abs/1904.12061.

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Notes

  1. 1.

    The concept of center is defined with respect to the graph instead of to the metric space.

  2. 2.

    This notation was used differently before. As we have several cases to consider, to save notation, we may use the same notation as long as the context is clear.

  3. 3.

    Note that \(i_2\le i_1+1\) always holds because \(d_P(v_1,v_{i_1+1})\ge d_P(v_{i_1+1},v_n)\ge |v_{i_1+1}v_n|\).

  4. 4.

    The index j must exist because \(d_P(v_1,v_i)\ge |v_iv_n|\) due to the definition of \(i_2\).

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Authors and Affiliations

  1. Department of Computer Science, Utah State University, Logan, UT, 84322, USA

    Christopher Johnson & Haitao Wang

Authors
  1. Christopher Johnson
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  2. Haitao Wang
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Correspondence to Haitao Wang .

Editor information

Editors and Affiliations

  1. Department of Computing Science, University of Alberta, Edmonton, AB, Canada

    Zachary Friggstad

  2. School of Computer Science, Carleton University, Manotick, ON, Canada

    Jörg-Rüdiger Sack

  3. Department of Computing Science, University of Alberta, Edmonton, AB, Canada

    Mohammad R Salavatipour

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Johnson, C., Wang, H. (2019). A Linear-Time Algorithm for Radius-Optimally Augmenting Paths in a Metric Space. In: Friggstad, Z., Sack, JR., Salavatipour, M. (eds) Algorithms and Data Structures. WADS 2019. Lecture Notes in Computer Science(), vol 11646. Springer, Cham. https://doi.org/10.1007/978-3-030-24766-9_34

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  • DOI: https://doi.org/10.1007/978-3-030-24766-9_34

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-24765-2

  • Online ISBN: 978-3-030-24766-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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