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

03/16/2023

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by   Timothy M. Chan, et al.

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We present a (combinatorial) algorithm with running time close to O(n^d) for computing the minimum directed L_∞ 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 Ω(n^d) for combinatorial algorithms, under the Combinatorial k-Clique Hypothesis.