Computing L_∞ Hausdorff Distances Under Translations: The Interplay of Dimensionality, Symmetry and Discreteness

Abstract

To measure the similarity of the shape of point sets, rather than their mere closeness in space, various notions of a Hausdorff distance under translation have been investigated. Specifically, let P and Q denote point sets of n and m points, respectively, in ℝ^d. We consider the task of computing the minimum distance d(P,Q+τ) over an admissible set of translations τ ∈ T, where d(⋅, ⋅) denotes the Hausdorff distance under the L_∞-norm. As variants, we distinguish between continuous (T = ℝ^d) or discrete (T is a given finite set of t translations) as well as directed or undirected (choosing the directed or undirected Hausdorff distance for d(⋅, ⋅)).
We seek to apply the paradigm of fine-grained complexity to understand the complexity of these variants, and in particular: How is the running time influenced by the dimension d, the relationship between n and m, and the specific choice of variant? As our main results, we obtain:  
- The asymmetric definition of the most studied variant, the continuous directed Hausdorff distance, results in an intrinsically asymmetric time complexity: While (Chan, SoCG'23) established a symmetric Õ((nm)^{d/2}) upper bound for all d ≥ 3 and proved it to be conditionally optimal for combinatorial algorithms whenever m ≤ n, we show that this lower bound does not hold for the case n ≪ m, by providing a combinatorial, almost-linear-time algorithm for d = 3 and n = m^{o(1)}. We further prove general, i.e., non-combinatorial, conditional lower bounds for d ≥ 3, in particular: (1) m^{⌊d/2⌋ - o(1)} for small n and (2) n^{d/2 - o(1)} for d = 3 and small m. 
- We observe that the directed and undirected case is closely related, in particular, all our lower bounds for d ≥ 3 hold for both the directed and undirected variant. A remarkable exception is the case of d = 1 for which we provide a conditional separation. Specifically, in contrast to the undirected variants being solvable in near-linear time (Rote, IPL'91), we show that the directed variants are at least as hard as the additive problem MaxConv LowerBound introduced in (Cygan, Mucha, Wegrzycki and Wlodarczyk, TALG'19). 
- We show that the discrete variants reduce to a variant of 3SUM for d ≤ 3. This gives a barrier in proving a tight lower bound of these variants under the Orthogonal Vectors Hypothesis (OVH); in contrast, the continuous variants admit a tight conditional lower bound under OVH in d = 2 (Bringmann, Nusser, JoCG'21).  These results reveal an intricate interplay of dimensionality, symmetry and discreteness in determining the fine-grained complexity of computing Hausdorff distances under translation.