Computing the Fr'echet Distance with a Retractable Leash

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 Rd under polyhedral distance functions (e.g., L1 and L_∞). We also get a (1+varepsilon)-approximation of the Fr'echet 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(n2 log2 n). However, we conjecture that it may eventually lead to a faster exact algorithm.