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

TL;DR

The results reveal an intricate interplay of dimensionality, symmetry, and discreteness in computing translational Hausdorff distances in computing translational Hausdorff distances.

Abstract

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.

Figures (7)

• Figure 1: Schematic figure of upper and lower bounds for directed Hausdorff under Translation in $3$-D (upper figure) and $d$-D for $d \geq 4$ (lower figure), novel results are marked with ($*$) . $\rho_1, \rho_2$ are functions on the ratio of $n,m$ which correspond to values described in \ref{['thm:lopsided-directed-LB']}.
• Figure 2: An example of an encoding of a $3$-partite graph to a $2$-D box (skewed cube). Normal lines represent edges, dotted lines represent a subset of non-edges. The gray boxes correspond to the forbidden regions of the respective non-edge. Note that in our reductions, we aim to cover the complement of these regions, the feasible regions.
• Figure 3: The reduction of a box in $1$-D, i.e., an interval $[a,b]$, to orthants in $2$-D. We prove that for each feasible translation $\tau'$ there exists a feasible translation $\tau'^*$ which lies on the diagonal $c(\mathbb{R})$. For that, we always assume that the translation $\tau '$ lies between the diagonal $c(\mathbb{R})$ and the vertex of the orthant, i.e., for this example the orthant of $c^-$ is relevant.
• Figure 4: An example of the feasible region of a non-edge $\bar{e} \in V_2 \times V_3 \times V_6 \setminus E$ covered by boxes, depicted in yellow, within a $d$-dimensional hypercube projected to $2$ dimensions.
• Figure 5: Left: The decomposition of a (skewed) hypercube in $2$-D into four identical sub-regions, marked in red, yellow, green, and blue. Boxes in equal color denote a common sub-region, here in each row there are two sub-regions that are split into two parts each. Right: The boxes used to cover the feasible region of the green sub-regions, in yellow, as well as the boxes covering the complement of the green sub-region, marked in pink.

Theorems & Definitions (52)

• Definition 1: Hausdorff under Translation (HuT)
• Definition 2: Discrete Hausdorff under Translation (DiscHuT)
• Definition 4: Translation Problem
• Definition 5: Translation Problem with Orthants (TPwO)
• Lemma 6
• Lemma 7
• Definition 8
• Lemma 9
• Lemma 11
• Lemma 12