Fine-Grained Analysis of Problems on Curves

We provide conditional lower bounds on two problems on polygonal curves. First, we generalize a recent result on the (discrete) Fréchet distance to k curves. Specifically, we show that, assuming the Strong Exponential Time Hypothesis, the Fréchet distance between k polygonal curves in the plane with n edges cannot be computed in O(nk−ε) time, for any ε > 0. Our second construction shows that under the same assumption a polygonal curve with n edges in dimension Ω(log n) cannot be simplified optimally in O(n2−ε) time.