Computing the Fréchet Gap Distance

Abstract

Measuring the similarity of two polygonal curves is a fundamental computational task. Among alternatives, the Fréchet distance is one of the most well-studied similarity measures. Informally, the Fréchet distance is described as the minimum leash length required for a man on one of the curves to walk a dog on the other curve continuously from the starting to the ending points. In this paper we study a variant called the Fréchet gap distance. In the man and dog analogy, the Fréchet gap distance minimizes the difference of the longest and smallest leash lengths used over the entire walk. This measure in some ways better captures our intuitive notions of curve similarity, for example giving distance zero to translated copies of the same curve. The Fréchet gap distance was originally introduced by Filtser and Katz [19] in the context of the discrete Fréchet distance. Here we study the continuous version, which presents a number of additional challenges not present in discrete case. In particular, the continuous nature makes bounding and searching over the critical events a rather difficult task. For this problem we give an $O(n^5\log n)$ time exact algorithm and a more efficient $O(n^2\log n +({n^2}/{{\varepsilon }})\log ({1}/{{\varepsilon }}))$ time $(1+{\varepsilon })$-approximation algorithm, where n is the total number of vertices of the input curves. Note that, ignoring logarithmic factors, for any constant ${\varepsilon }$ our approximation has quadratic running time, matching the lower bound, assuming SETH [Bringmann, K.: Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless SETH fails (2014)], for approximating the standard Fréchet distance for general curves.