Adaptive Computation of the Discrete Fr\'echet Distance

Abstract

The discrete Fr{\'e}chet distance is a measure of similarity between point sequences which permits to abstract differences of resolution between the two curves, approximating the original Fr{\'e}chet distance between curves. Such distance between sequences of respective length $n$ and $m$ can be computed in time within $O(nm)$ and space within $O(n+m)$ using classical dynamic programing techniques, a complexity likely to be optimal in the worst case over sequences of similar lenght unless the Strong Exponential Hypothesis is proved incorrect. We propose a parameterized analysis of the computational complexity of the discrete Fr{\'e}chet distance in fonction of the area of the dynamic program matrix relevant to the computation, measured by its \emph{certificate width} $\omega$. We prove that the discrete Fr{\'e}chet distance can be computed in time within $((n+m)\omega)$ and space within $O(n+m+\omega)$.