Computing the ~%DIFDELCMD < \Frd%%% \ 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 %DIFDELCMD < \Reals%%%  ^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 %DIFDELCMD < \Frd%%% \ 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.
Computing the %DIFDELCMD < \Frd%%% 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 %DIFDELCMD < \Reals%%% \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 %DIFDELCMD < \Frd%%% Fr\'echet 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.
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Fr\'echet distance
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Fr\'echet distance