Approximating Metric Spaces by Tree Metrics

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Keywords and Synonyms

Embedding general metrics into tree metrics     

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 Cn 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 Cn approximates the distances between any two nodes in Cn 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 ...