Computing the Fréchet Distance with a Retractable Leash

Abstract

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 Rd under polyhedral distance functions (e.g., L1 and L∞). We also get a (1 + ε) -approximation of the Fréchet distance under the Euclidean metric, in quadratic time for any fixed ε> 0. For the exact Euclidean case, our framework currently yields an algorithm with running time O(n2log 2n). However, we conjecture that it may eventually lead to a faster exact algorithm.