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Computing the Fréchet Distance with a Retractable Leash

All known algorithms for the Fréchet 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 Fréchet distance between polygonal curves in $R^d$ under polyhedral distance functions (e.g., $L_1$ and $L_\infty$). We also get a $(1+\varepsilon)$-approximation of the Fréchet distance under the Euclidean metric, in quadratic time for any fixed $\varepsilon > 0$. For the exact Euclidean case, our framework currently yields an algorithm with running time $O(n^2 \log^2 n)$. However, we conjecture that it may eventually lead to a faster exact algorithm.