Computing the Gromov hyperbolicity of a discrete metric space
Abstract
We give exact and approximation algorithms for computing the Gromov hyperbolicity of an n-point discrete metric space. We observe that computing the Gromov hyperbolicity from a fixed base-point reduces to a (max,min) matrix product. Hence, using the (max,min) matrix product algorithm by Duan and Pettie, the fixed base-point hyperbolicity can be determined in O(n^2.69) time. It follows that the Gromov hyperbolicity can be computed in O(n^3.69) time, and a 2-approximation can be found in O(n^2.69) time. We also give a (2 log_2 n)-approximation algorithm that runs in O(n^2) time, based on a tree-metric embedding by Gromov. We also show that hyperbolicity at a fixed base-point cannot be computed in O(n^2.05) time, unless there exists a faster algorithm for (max,min) matrix multiplication than currently known.
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Cite
@article{arxiv.1210.3323,
title = {Computing the Gromov hyperbolicity of a discrete metric space},
author = {Hervé Fournier and Anas Ismail and Antoine Vigneron},
journal= {arXiv preprint arXiv:1210.3323},
year = {2015}
}
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6 pages