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arXiv · 1509.05751

Computing the Gromov-Hausdorff Distance for Metric Trees

Pankaj K. Agarwal·Kyle Fox·Abhinandan Nath·Anastasios Sidiropoulos·Yusu Wang
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Abstract

The Gromov-Hausdorff (GH) distance is a natural way to measure distance between two metric spaces. We prove that it is $\mathrm{NP}$-hard to approximate the Gromov-Hausdorff distance better than a factor of $3$ for geodesic metrics on a pair of trees. We complement this result by providing a polynomial time $O(\min\{n, \sqrt{rn}\})$-approximation algorithm for computing the GH distance between a pair of metric trees, where $r$ is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an $O(\sqrt{n})$-approximation algorithm.

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Pankaj K. Agarwal, Kyle Fox, Abhinandan Nath, Anastasios Sidiropoulos, Yusu Wang. 2015-09-18. Computing the Gromov-Hausdorff Distance for Metric Trees. https://arxiv.org/abs/1509.05751

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