Computing the Fréchet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fréchet distance computations become easier. Let P and Q be two polygonal curves in 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 Fréchet distance between them: (1) a linear-time algorithm for deciding the Fréchet distance between two curves, (2) an algorithm that computes the Fréchet distance in O((n + m) log(n + m)) time, (3) a linear-time √ 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. 23:2Fast Fréchet Distance Between Curves with Long Edges O(n 2 /φ + n log n)-time algorithm to compute a φ-approximation of the discrete Fréchet distance for any integer 1 ≤ φ ≤ n. Therefore, an n -approximation, for any > 0, can be computed in (strongly) subquadratic time. For the continuous Fréchet distance, there are also a few subquadratic algorithms known for restricted classes of curves such as κ-bounded, backbone and c-packed curves. Alt et al. [3] considered κ-bounded curves and they gave an O(n log n) time algorithm to (κ + 1)-approximate the Fréchet distance. A curve P is κbounded if for any two points x, y ∈ P , the union of the balls with radii r centered at x and y contains the whole P [x, y] where r is equal to (κ/2) times the Euclidean distance between x and y. For any > 0, Aronov et al. [4] provided a near-linear time (1 + )-approximation algorithm for the discrete Fréchet distance for so-called backbone curves that have essentially constant edge length and require a minimum distance between non-consecutive vertices. For c-packed curves a (1 + )-approximation can be computed in O(cn/ + cn log n) time [11]. A curve is c-packed if for any ball B, the length of the portion of P contained in B is at most c times the diameter of B.

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