We present a (combinatorial) algorithm with running time close to $O(n^d)$ for computing the minimum directed $L_\infty$ Hausdorff distance between two sets of $n$ points under translations in any constant dimension $d$. This substantially improves the best previous time bound near $O(n^{5d/4})$ by Chew, Dor, Efrat, and Kedem from more than twenty years ago. Our solution is obtained by a new generalization of Chan's algorithm [FOCS'13] for Klee's measure problem. To complement this algorithmic result, we also prove a nearly matching conditional lower bound close to $\Omega(n^d)$ for combinatorial algorithms, under the Combinatorial $k$-Clique Hypothesis.

翻译：我们提出了一种运行时间接近$O(n^d)$的（组合）算法，用于计算任意常数维度$d$下两个大小为$n$的点集在平移变换下的最小有向$L_\infty$豪斯多夫距离。这显著改进了二十余年前Chew、Dor、Efrat和Kedem给出的近$O(n^{5d/4})$的最佳先前时间界。我们的解法通过推广Chan在FOCS'13上针对Klee测度问题的算法得到。为补充这一算法结果，我们还证明了在组合k-团假设下，组合算法几乎匹配的条件下界接近$\Omega(n^d)$。