Computing the Fréchet Distance with a Retractable Leash

Kevin Buchin, Maike Buchin, Rolf van Leusden, Wouter Meulemans, Wolfgang Mulzer
2016 Discrete & Computational Geometry  
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_∞). We also get a
more » ... pproximation 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(n^2 ^2 n). However, we conjecture that it may eventually lead to a faster exact algorithm.
doi:10.1007/s00454-016-9800-8 fatcat:okpbqvy7tvgqvlqbr76jimuyli