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We present a (combinatorial) algorithm with running time close to $O(n^d)$for computing the minimum directed $L_\infty$ Hausdorff distance between twosets of $n$ points under translations in any constant dimension $d$. Thissubstantially 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 obtainedby a new generalization of Chan's algorithm [FOCS'13] for Klee's measureproblem.  To complement this algorithmic result, we also prove a nearly matchingconditional lower bound close to $\Omega(n^d)$ for combinatorial algorithms,under the Combinatorial $k$-Clique Hypothesis.

Minimum $L_\infty$ Hausdorff Distance of Point Sets Under Translation: Generalizing Klee's Measure Problem