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All known algorithms for the Fr\'echet distance between curves proceed in twosteps: first, they construct an efficient oracle for the decision version;second, they use this oracle to find the optimum from a finite set of criticalvalues. We present a novel approach that avoids the detour through the decisionversion. This gives the first quadratic time algorithm for the Fr\'echetdistance between polygonal curves in $R^d$ under polyhedral distance functions(e.g., $L_1$ and $L_\infty$). We also get a $(1+\varepsilon)$-approximation ofthe Fr\'echet distance under the Euclidean metric, in quadratic time for anyfixed $\varepsilon > 0$. For the exact Euclidean case, our framework currentlyyields an algorithm with running time $O(n^2 \log^2 n)$. However, we conjecturethat it may eventually lead to a faster exact algorithm.

Computing the Fr\'echet Distance with a Retractable Leash