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Fast Approximation Algorithms for p-Centers in Large \(\delta \)-Hyperbolic Graphs

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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 \(O(p(\delta +1)(n+m)\log (n))\) 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 \(O((n^3\log n + n^2m)\log ({{\mathrm{diam}}}(G)))\) which is impractical for large graphs.

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Notes

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

  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 [4] for details.

  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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Authors and Affiliations

  1. Mathematics and Algorithms Research, Bell Labs, Nokia, 600 Mountain Avenue, Murray Hill, NJ, 07974, USA

    Katherine Edwards & Iraj Saniee

  2. Mathematics and Algorithms Research, Bell Labs, Nokia, 600 March Road, Ottawa, ON, K2K2T6, Canada

    W. Sean Kennedy

Authors
  1. Katherine Edwards
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  2. W. Sean Kennedy
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  3. Iraj Saniee
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Correspondence to Katherine Edwards.

Additional information

Katherine Edwards’s research was partially supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 279558.

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Edwards, K., Kennedy, W.S. & Saniee, I. Fast Approximation Algorithms for p-Centers in Large \(\delta \)-Hyperbolic Graphs. Algorithmica 80, 3889–3907 (2018). https://doi.org/10.1007/s00453-018-0425-6

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  • DOI: https://doi.org/10.1007/s00453-018-0425-6

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