Translating Hausdorff is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation

TL;DR

This paper establishes tight computational lower bounds for calculating the Hausdorff distance under translation, revealing the problem's inherent difficulty and matching the best known algorithms' complexity bounds.

Contribution

It provides the first fine-grained complexity lower bounds for Hausdorff distance under translation, matching existing upper bounds under standard hypotheses.

Findings

Matching lower bounds of $(nm)^{1-o(1)}$ for $L_1$ and $L_ ext{infinity}$ norms.

Matching lower bound of $n^{2-o(1)}$ for $L_2$ norm in the imbalanced case.

Results are based on the Orthogonal Vectors and 3SUM hypotheses.

Abstract

Computing the similarity of two point sets is a 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 $L_\infty$ (and all other $L_p$ norms) 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.