Computing the similarity of two point sets is an ubiquitous task in medical imaging, geometric shape comparison, trajectory analysis, and many more settings. Arguably the most basic distance measure for this task is the Hausdorff distance, which assigns to each point from one set the closest point in the other set and then evaluates the maximum distance of any assigned pair. A drawback is that this distance measure is not translational invariant, that is, comparing two objects just according to their shape while disregarding their position in space is impossible. Fortunately, there is a canonical translational invariant version, the Hausdorff distance under translation, which minimizes the Hausdorff distance over all translations of one of the point sets. For point sets of size $n$ and $m$, the Hausdorff distance under translation can be computed in time $\tilde O(nm)$ for the $L_1$ and $L_\inf$ norm [Chew, Kedem SWAT'92] and $\tilde O(nm (n+m))$ for the $L_2$ norm [Huttenlocher, Kedem, Sharir DCG'93]. As these bounds have not been improved for over 25 years, in this paper we approach the Hausdorff distance under translation from the perspective of fine-grained complexity theory. We show (i) a matching lower bound of $(nm)^{1-o(1)}$ for $L_1$ and $L_\inf$ assuming the Orthogonal Vectors Hypothesis and (ii) a matching lower bound of $n^{2-o(1)}$ for $L_2$ in the imbalanced case of $m = O(1)$ assuming the 3SUM Hypothesis.

翻译：在医学成像、几何形状比较、轨迹分析以及更多设置中,计算两个点的相似度值2 是一个无处不在的任务。 任务的最基本距离量度是Hausdorf距离, 它从另一组中设定一个最接近的点指派给每个点, 然后评价任何指定对配的最大距离。 一个缺点是, 这个距离量度不是翻译性的, 也就是说, 在忽略其在空间的位置时, 将两个物体按其形状比较 [Chew, Kedem SWAT'92] 标准。 幸运的是, 翻译中存在一个罐头翻译版本, 豪斯多夫距离, 将所有点各组的翻译中的豪斯多夫距离最小化。 对于一个大小为$和美元, 翻译中的豪斯多夫距离可以用时间来计算 $tilde O(nm) 美元和 美元( 美元) 标准 [Chdemode, 在 25G2 标准下, 在硬页中显示一个硬质的硬值 1 。