  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.
