SETH Says: Weak Fr\'echet Distance is Faster, but only if it is Continuous and in One Dimension

TL;DR

This paper establishes tight computational bounds for approximating the Fréchet distance, showing that in one dimension, the weak Fréchet distance can be computed exactly in linear time, unlike the continuous case.

Contribution

It proves strong lower bounds for approximation of the Fréchet distance and provides a linear-time algorithm for the weak Fréchet distance in one dimension.

Findings

Strong SETH-based lower bounds for approximation in higher dimensions.

Exact linear-time algorithm for weak Fréchet distance in one dimension.

Different behaviors of continuous and weak Fréchet distances in one dimension.

Abstract

We show by reduction from the Orthogonal Vectors problem that algorithms with strongly subquadratic running time cannot approximate the Fr\'echet distance between curves better than a factor $3$ unless SETH fails. We show that similar reductions cannot achieve a lower bound with a factor better than $3$. Our lower bound holds for the continuous, the discrete, and the weak discrete Fr\'echet distance even for curves in one dimension. Interestingly, the continuous weak Fr\'echet distance behaves differently. Our lower bound still holds for curves in two dimensions and higher. However, for curves in one dimension, we provide an exact algorithm to compute the weak Fr\'echet distance in linear time.