Computing the similarity of two point sets is a 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 $L_\infty$ 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 $L_\infty$ (and all other $L_p$ norms) 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,计算两个点的相似性是一个无处不在的任务。 幸运的是, 此任务最基本的距离测量标准是Hausdorf距离, 它从另一组中设定一个最接近的点指派给每个点, 然后评估任何指定配对的最大距离。 一个缺点是, 此距离测量标准不是翻译性的, 也就是说, 比较两个对象时, 忽略其在空间的位置是不可能做到的。 幸运的是, 存在一个罐头翻译的变异版本, 翻译的Hausdorf距离, 将Hausdorf距离从另一个最接近的点指定点指派到另一个最接近的点, 美元=O(nm), 也就是说, 美元和 美元( K) 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 折价, 美元, 美元, 美元, 。, 。