Conjecture: O(nm log(nm)) algorithm for continuous barking distance

Develop an algorithm that, given two polygonal curves P (with n vertices) and Q (with m vertices) in a fixed-dimensional Euclidean space, a barking radius ρ > 0, and a speed bound s > 0, computes the continuous barking distance from P to Q in O(nm log(nm)) time. The continuous barking distance is defined as the minimum, over all differentiable one-sided reparametrizations γ:[0,|P|]→[0,|Q|] with |γ'(t)| ≤ s, of the integral ∫₀^{|P|} 1{d(P[t], Q[γ(t)]) > ρ} dt. Establish such an algorithm that matches the near-quadratic time suggested by the conjecture.

Background

The paper introduces the barking distance, an asymmetric similarity measure designed for detecting outliers between curves, and provides algorithms for discrete and semi-discrete variants that run in near-optimal time (O(nm log s) and O(nm log(nm)), respectively), along with SETH-based lower bounds ruling out truly subquadratic algorithms for all settings.

For the continuous setting, the authors present a polynomial-time algorithm with running time O(n^4 m^3 log(nm)) and suggest that techniques similar to those used in the semi-discrete case might improve this to O(n m^3 log m). However, a substantial gap remains between this and the near-quadratic time, motivating a conjecture that an O(nm log(nm)) algorithm exists for the continuous barking distance.

This conjecture seeks an algorithm whose time complexity matches the SETH-consistent near-quadratic barrier, aligning the continuous case with the discrete and semi-discrete variants in terms of computational efficiency.