Computing the Fréchet Distance with a Retractable Leash
Kevin Buchin, Maike Buchin, Rolf van Leusden, Wouter Meulemans, Wolfgang Mulzer · Discrete & Computational Geometry · 2016
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 $$\mathbb {R}^d$$ 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.