度量树的Gromov-Hausdorff距离计算
Computing the Gromov-Hausdorff Distance for Metric Trees
摘要：
Gromov-Hausdorff（GH）距离是度量两个度量空间之间距离的一种自然方法。我们证明了它是$\mathrm{NP}$ -难以近似的Gromov-Hausdorff距离比一个因子$3$的测地线度量对树。我们补充这一结果，提供了一个多项式时间$O(\min\{n, \sqrt{rn}\})$近似算法计算GH之间的距离对度量树，其中$r$是在两棵树的最长边长度的比率，最短的边长度。对于具有单位长度边的度量树，这产生$O(\sqrt{n})$ -近似算法。
英文摘要：
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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