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Approximating Metric Spaces by Tree Metrics

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  • First Online:
Encyclopedia of Algorithms

Years and Authors of Summarized Original Work

  • 1996; Bartal, Fakcharoenphol, Rao, Talwar

  • 2004; Bartal, Fakcharoenphol, Rao, Talwar

Problem Definition

This problem is to construct a random tree metric that probabilistically approximates a given arbitrary metric well. A solution to this problem is useful as the first step for numerous approximation algorithms because usually solving problems on trees is easier than on general graphs. It also finds applications in on-line and distributed computation.

It is known that tree metrics approximate general metrics badly, e.g., given a cycle C n with n nodes, any tree metric approximating this graph metric has distortion \( { \Omega(n) } \) [17]. However, Karp [15] noticed that a random spanning tree of C n approximates the distances between any two nodes in C n well in expectation. Alon, Karp, Peleg, and West [1] then proved a bound of \( { \exp(O(\sqrt{\log n\log\log n})) } \)on an average distortion for approximating any graph metric with its...

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Recommended Reading

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Fakcharoenphol, J., Rao, S., Talwar, K. (2016). Approximating Metric Spaces by Tree Metrics. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_25

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