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Making Doubling Metrics Geodesic

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Abstract

The starting point of our research is the following problem: given a doubling metric ℳ=(V,d), can one (efficiently) find an unweighted graph G′=(V′,E′) with V⊆V′ whose shortest-path metric d′ is still doubling, and which agrees with d on V×V? While it is simple to show that the answer to the above question is negative if distances must be preserved exactly. However, allowing a (1+ε) distortion between d and d′ enables us bypass this hurdle, and obtain an unweighted graph G′ with doubling dimension at most a factor O(log ε −1) times the doubling dimension of G.

More generally, this paper gives algorithms that construct graphs G′ whose convex (or geodesic) closure has doubling dimension close to that of ℳ, and the shortest-path distances in G′ closely approximate those of ℳ when restricted to V×V. Similar results are shown when the metric ℳ is an additive (tree) metric and the graph G′ is restricted to be a tree.

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Authors and Affiliations

  1. Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Anupam Gupta

  2. Microsoft Research, Silicon Valley Campus, 1065 La Avenida, Mountain View, CA, 94043, USA

    Kunal Talwar

Authors
  1. Anupam Gupta
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  2. Kunal Talwar
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Corresponding author

Correspondence to Kunal Talwar.

Additional information

This research was partly supported by the NSF CAREER award CCF-0448095 and CCF-0729022, and by an Alfred P. Sloan Fellowship.

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Gupta, A., Talwar, K. Making Doubling Metrics Geodesic. Algorithmica 59, 66–80 (2011). https://doi.org/10.1007/s00453-010-9397-x

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  • DOI: https://doi.org/10.1007/s00453-010-9397-x

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