Jaywalking your dog: Computing the Fréchet distance with shortcuts

The similarity of two polygonal curves can be measured using the Fréchet distance. We introduce the notion of a more robust Fréchet distance, where one is allowed to shortcut between vertices of one of the curves. This is a natural approach for handling noise, in particular batched outliers. We compute a constant factor approximation to the minimum Fréchet distance over all possible such shortcuts. Our algorithm runs in O(c^2 kn log^3 n) time if one is allowed to take at most k shortcuts and the input curves are c-packed. For the case where the number of shortcuts is unrestricted, we describe an algorithm which runs in O(c^2 n log^3 n) time. To facilitate the new algorithm we develop several new data-structures, which we believe to be of independent interest: (i) for range reporting on a curve, and (ii) for preprocessing a curve to answer queries for the Fréchet distance between a subcurve and a line segment.