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Fast Approximation Algorithms for p-centers in Large \(\delta \)-hyperbolic Graphs

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Algorithms and Models for the Web Graph (WAW 2016)

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Abstract

We provide a quasilinear time algorithm for the p-center problem with an additive error less than or equal to 3 times the input graph’s hyperbolic constant. Specifically, for the graph \(G=(V,E)\) with n vertices, m edges and hyperbolic constant \(\delta \), we construct an algorithm for p-centers in time \(O(p(\delta +1)(n+m)\log _2(n))\) with radius not exceeding \(r_p + \delta \) when \(p \le 2\) and \(r_p + 3\delta \) when \(p \ge 3\), where \(r_p\) are the optimal radii. Prior work identified p-centers with accuracy \(r_p+\delta \) but with time complexity \(O((n^3\log _2 n + n^2m)\log _2({{\mathrm{diam}}}(G)))\) which is impractical for large graphs.

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Notes

  1. 1.

    For a comprehensive treatment of \(\delta \)-hyperbolicity see [1].

  2. 2.

    The cited result also gives rise to an algorithm for general \(\delta \)-hyperbolic spaces whose running time depends on the time to compute \(F_S(x)\) for \(x\in X\) and \(S\subseteq X\). Because our interest is primarily in graphs, we direct the reader to [3] for details.

  3. 3.

    The graphs p2p-gnutella25 and web-stanford are available publicly as part of the Stanford Large Network Dataset Collection. The sn-medium graph is extracted from the social network Facebook, and the sprintlink-1239 graph is an IP-layer network from the Rocketfuel ISP.

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Acknowledgement

The authors are grateful to the anonymous reviewers for helpful comments.

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Authors and Affiliations

  1. Department of Computer Science, Princeton University, Princeton, NJ, 08540, USA

    Katherine Edwards

  2. Mathematics of Networks and Communications Department, Bell Labs, Nokia, Holmdel, NJ, 07974, USA

    Sean Kennedy & Iraj Saniee

Authors
  1. Katherine Edwards
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  2. Sean Kennedy
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  3. Iraj Saniee
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Corresponding author

Correspondence to Katherine Edwards .

Editor information

Editors and Affiliations

  1. Ryerson University, Toronto, Canada

    Anthony Bonato

  2. University of California San Diego, La Jolla, California, USA

    Fan Chung Graham

  3. Ryerson University, Toronto, Canada

    Paweł Prałat

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Edwards, K., Kennedy, S., Saniee, I. (2016). Fast Approximation Algorithms for p-centers in Large \(\delta \)-hyperbolic Graphs. In: Bonato, A., Graham, F., Prałat, P. (eds) Algorithms and Models for the Web Graph. WAW 2016. Lecture Notes in Computer Science(), vol 10088. Springer, Cham. https://doi.org/10.1007/978-3-319-49787-7_6

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  • DOI: https://doi.org/10.1007/978-3-319-49787-7_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-49786-0

  • Online ISBN: 978-3-319-49787-7

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