Computing the Fréchet Distance with a Retractable Leash
Article, Preprint, Report
van Leusden, R.
- Publisher: Springer Nature
Discrete & Computational Geometry
Theoretical Computer Science | Geometry and Topology | Computational Theory and Mathematics | Computer Science - Computational Geometry | QA75 | Discrete Mathematics and Combinatorics
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 (Formula presented.) under polyhedral distance functions (e.g., (Formula presented.) and (Formula presented.)). We also get a (Formula presented.)-approximation of the Fréchet distance under the Euclidean metric, in quadratic time for any fixed (Formula presented.). For the exact Euclidean case, our framework currently yields an algorithm with running time (Formula presented.). However, we conjecture that it may eventually lead to a faster exact algorithm.