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

2026-03-09

AI Summary

This study investigates the computational complexity of computing the $L_\infty$ Hausdorff distance between point sets under translation, distinguishing between continuous versus discrete translations and directed versus undirected settings. By integrating fine-grained complexity hypotheses—such as MaxConv, 3SUM, and the Orthogonal Vectors Hypothesis (OVH)—with combinatorial algorithm design, the work elucidates the interplay among dimensionality, symmetry, and discreteness. Key contributions include an almost-linear-time algorithm for $d=3$ when $n = m^{o(1)}$, conditional lower bounds for multiple variants in dimensions $d \geq 3$, a reduction showing that the discrete case for $d \leq 3$ is 3SUM-hard—which limits the tightness of OVH-based lower bounds—and a proof that the directed problem in $d=1$ is MaxConv-hard, thereby establishing the first complexity separation between directed and undirected formulations.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• Hausdorff distance
• L∞ norm
• translations
• fine-grained complexity
• dimensionality

Innovation

• Hausdorff distance
• fine-grained complexity
• asymmetric time complexity
• MaxConv hardness
• 3SUM reduction