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