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

Abstract

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.

To appear in SoCG 2023