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 $d(P,Q+τ)$ over translations $τ\in T$, 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.

翻译：为度量点集的形状相似性，平移下的Hausdorff距离的多种概念被广泛研究。在此背景下，对于$\mathbb{R}^d$中的$n$点集$P$和$m$点集$Q$，我们考虑计算平移$τ\in T$下的最小$d(P,Q+τ)$的任务，其中$d(\cdot, \cdot)$表示$L_\infty$范数下的Hausdorff距离。我们分析了连续（$T=\mathbb{R}^d$）与离散（$T$为有限集）以及有向与无向的变体。应用细粒度复杂性理论，我们分析了运行时间对维度$d$、$n$与$m$的关系以及所选变体的依赖性。我们的主要结果是：（1）连续有向Hausdorff距离具有非对称时间复杂度。虽然（Chan, SoCG'23）对$d\ge 3$给出了对称的$\tilde{O}((nm)^{d/2})$上界，且当$m \le n$时该界对于组合算法是条件最优的，但我们证明当$n \ll m$时该界不成立，并针对$d=3$和$n=m^{o(1)}$给出了一个组合的、几乎线性时间的算法。我们还证明了$d\ge 3$时的一般条件性下界：对于小的$n$为$m^{\lfloor d/2 \rfloor -o(1)}$，对于$d=3$和小的$m$为$n^{d/2 -o(1)}$。（2）虽然$d \ge 3$时的下界对有向和无向变体均成立，但$d=1$时产生了条件性分离。与可在近线性时间内求解的无向变体（Rote, IPL'91）不同，我们证明有向变体至少与加法MaxConv下界问题（Cygan et al., TALG'19）一样困难。（3）对于$d\le 3$，离散变体可归约为一个3SUM变体。这为在正交向量假设（OVH）下证明紧下界设置了障碍，与连续变体在$d=2$时允许基于OVH的紧下界（Bringmann, Nusser, JoCG'21）形成对比。这些结果揭示了在计算平移Hausdorff距离时维度、对称性与离散性之间复杂的相互作用。