Abstract
Tree embeddings are a powerful tool in the area of graph approximation algorithms. Essentially, they transform problems on general graphs into much easier ones on trees. Fakcharoenphol, Rao, and Talwar (FRT) [STOC’04] present a probabilistic tree embedding that transforms n-node metrics into (probability distributions over) trees, while stretching each pairwise distance by at most an O(logn) factor in expectation. This O(logn) stretch is optimal.
Khan et al. [PODC’08] present a distributed algorithm that implements FRT in O(SPD logn) rounds, where SPD is the shortest-path-diameter of the weighted graph, and they explain how to use this embedding for various distributed approximation problems. Note that SPD can be as large as Θ(n), even in graphs where the hop-diameter D is a constant. Khan et al. noted that it would be interesting to improve this complexity. We show that this is indeed possible.
More precisely, we present a distributed algorithm that constructs a tree embedding that is essentially as good as FRT in \(\tilde{O}(\min\{n^{0.5+\varepsilon },\operatorname{SPD}\}+D)\) rounds, for any constant ε > 0. A lower bound of \(\tilde{\Omega}(\min\{n^{0.5},\operatorname{SPD}\}+D)\) rounds follows from Das Sarma et al. [STOC’11], rendering our round complexity near-optimal.
This work was supported by AFOSR contract number FA9550-13-1-0042, NSF award 0939370-CCF, NSF award CCF-1217506, NSF award CCF-AF-0937274, and DFG funding Le 3107/1-1. The first author is also thankful for the support of Simons Award for graduate students in Theoretical Computer Science (number 31872).
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Ghaffari, M., Lenzen, C. (2014). Near-Optimal Distributed Tree Embedding. In: Kuhn, F. (eds) Distributed Computing. DISC 2014. Lecture Notes in Computer Science, vol 8784. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45174-8_14
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DOI: https://doi.org/10.1007/978-3-662-45174-8_14
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