Algorithmica

, Volume 59, Issue 1, pp 66–80

Making Doubling Metrics Geodesic

Authors

  • Anupam Gupta
    • Computer Science DepartmentCarnegie Mellon University
    • Microsoft Research
Article

DOI: 10.1007/s00453-010-9397-x

Cite this article as:
Gupta, A. & Talwar, K. Algorithmica (2011) 59: 66. doi:10.1007/s00453-010-9397-x
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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 VV′ 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.

Keywords

Metric embeddingsLow-distortion embeddingsDoubling metricsGeodesic metricsConvex closure

Copyright information

© Springer Science+Business Media, LLC 2010