Computing $L_\infty$ Hausdorff Distances Under Translations: The Interplay of Dimensionality, Symmetry and Discreteness

To measure the shape similarity of point sets, various notions of the Hausdorff distance under translation are widely studied. In this context, for an $n$-point set $P$ and $m$-point set $Q$ in $\mathbb{R}^d$, we consider the task of computing the minimum $d(P,Q+\tau)$ over translations $\tau \in T$, where $d(\cdot, \cdot)$ denotes the Hausdorff distance under the $L_\infty$-norm. We analyze continuous ($T=\mathbb{R}^d$) vs. discrete ($T$ is finite) and directed vs. undirected variants. Applying fine-grained complexity, we analyze running time dependencies on dimension $d$, the $n$ vs. $m$ relationship, and the chosen variant. Our main results are: (1) The continuous directed Hausdorff distance has asymmetric time complexity. While (Chan, SoCG'23) gave a symmetric $\tilde{O}((nm)^{d/2})$ upper bound for $d\ge 3$, which is conditionally optimal for combinatorial algorithms when $m \le n$, we show this fails for $n \ll m$ with a combinatorial, almost-linear time algorithm for $d=3$ and $n=m^{o(1)}$. We also prove general conditional lower bounds for $d\ge 3$: $m^{\lfloor d/2 \rfloor -o(1)}$ for small $n$, and $n^{d/2 -o(1)}$ for $d=3$ and small $m$. (2) While lower bounds for $d \ge 3$ hold for directed and undirected variants, $d=1$ yields a conditional separation. Unlike undirected variants solvable in near-linear time (Rote, IPL'91), we prove directed variants are at least as hard as the additive MaxConv LowerBound (Cygan et al., TALG'19). (3) The discrete variant reduces to a 3SUM variant for $d\le 3$. This creates a barrier to proving tight lower bounds under the Orthogonal Vectors Hypothesis (OVH), contrasting with continuous variants that admit tight OVH-based lower bounds in $d=2$ (Bringmann, Nusser, JoCG'21). These results reveal an intricate interplay of dimensionality, symmetry, and discreteness in computing translational Hausdorff distances.