The Discrete Fréchet Gap

We introduce the discrete Fr\'echet gap and its variants as an alternative measure of similarity between polygonal curves. We believe that for some applications the new measure (and its variants) may better reflect our intuitive notion of similarity than the discrete Fr\'echet distance (and its variants), since the latter measure is indifferent to (matched) pairs of points that are relatively close to each other. Referring to the frogs analogy by which the discrete Fr\'echet distance is often described, the discrete Fr\'echet gap is the minimum difference between the longest and shortest positions of the leash needed for the frogs to traverse their point sequences. We present an optimization scheme, which is suitable for any monotone function defined for pairs of distances such as the gap and ratio functions. We apply this scheme to two variants of the discrete Fr\'echet gap, namely, the one-sided discrete Fr\'echet gap with shortcuts and the weak discrete Fr\'echet gap, to obtain $O(n^2 \log^2 n)$-time algorithms for computing them.