The Gromov-Hausdorff distance is a natural way to measure distance between two metric spaces. We give the first proof of hardness and first non-trivial approximation algorithm for computing the Gromov-Hausdorff distance for geodesic metrics in trees. Specifically, we prove it is NP-hard to approximate the Gromov-Hausdorff distance better than a factor of 3. We complement this result by providing a polynomial time O(min{n, √rn})-approximation algorithm 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(√ n)-approximation algorithm.
CITATION STYLE
Agarwal, P. K., Fox, K., Nath, A., Sidiropoulos, A., & Wang, Y. (2015). Computing the Gromov-Hausdorff distance for metric trees. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 9472, pp. 529–540). Springer Verlag. https://doi.org/10.1007/978-3-662-48971-0_45
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