  Computing the Fr\'echet distance between two polygonal curves takes roughly
quadratic time. In this paper, we show that for a special class of curves the
Fr\'echet distance computations become easier. Let $P$ and $Q$ be two polygonal
curves in $\mathbb{R}^d$ with $n$ and $m$ vertices, respectively. We prove four
results for the case when all edges of both curves are long compared to the
Fr\'echet distance between them: (1) a linear-time algorithm for deciding the
Fr\'echet distance between two curves, (2) an algorithm that computes the
Fr\'echet distance in $O((n+m)\log (n+m))$ time, (3) a linear-time
$\sqrt{d}$-approximation algorithm, and (4) a data structure that supports
$O(m\log^2 n)$-time decision queries, where $m$ is the number of vertices of
the query curve and $n$ the number of vertices of the preprocessed curve.
