Computing the similarity of two point sets is an 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 _\inf 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 _\inf 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.
Computing the similarity of two point sets is an 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 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.
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