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

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.