Computing the Fréchet Distance with a Retractable Leash

Abstract: All known algorithms for the Fr\'echet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fr\'echet distance between polygonal curves in $R<sup>d$ under polyhedral distance functions (e.g., $L_1$ and $L_\infty$). We also get a $(1+\varepsilon)$-approximation of the Fr\'echet distance under the Euclidean metric, in quadratic time for any fixed $\varepsilon &gt; 0$. For the exact Euclidean case, our framework currently yields an algorithm with running time $O(n² \log² n)$. However, we conjecture that it may eventually lead to a faster exact algorithm.