We present the prefix Fréchet similarity as a new measure for similarity of curves which is for instance motivated by evacuation analysis and defined as follows. Given two (polygonal) curves T and $T^{\prime }$, we ask for two prefix curves of T and $T^{\prime }$ which have a Fréchet distance no larger than a given distance threshold $\delta \ge 0$ with respect to the $L_1$ metric such that the sum of the lengths of the prefix curves is maximal. As parameterized Fréchet measures as for example the prefix Fréchet similarity are highly unstable regarding the value of the distance threshold δ, we give an algorithm that computes exactly the profile of the prefix Fréchet similarity which means the complete functional relation between δ and the prefix Fréchet similarity of T and $T^{\prime }$. This is the first efficient algorithm for computing exactly the whole profile of a parametrized Fréchet distance.

While the running time of our algorithm for computing the profile of the prefix Fréchet similarity is $\mathcal{O}(n^3\log n)$, we provide a lower bound of $\mathrm{\Omega }(n^2)$ for the running time of any algorithm computing the profile of the prefix Fréchet similarity, where n denotes the number of segments on T and $T^{\prime }$. This implies that our running time is at most a near linear factor away from being optimal.

我们将前缀 Fréchet 相似性作为曲线相似性的新度量，例如由疏散分析激发并定义如下。给定两条（多边形）曲线T和${吨}^{\prime }$我们问了两个前缀曲线的牛逼和${吨}^{\prime }$ 其 Fréchet 距离不大于给定的距离阈值 $\delta \ge 0$ 相对于该 ${升}_1$度量使得前缀曲线的长度之和最大。由于参数化的 Fréchet 度量，例如前缀 Fréchet 相似度对于距离阈值δ的值非常不稳定，我们给出了一种算法来精确计算前缀 Fréchet 相似度的轮廓，这意味着δ和前缀 Fréchet 相似度之间的完整函数关系的Ť和${吨}^{\prime }$. 这是用于精确计算参数化 Fréchet 距离的整个轮廓的第一个有效算法。

而我们计算前缀 Fréchet 相似度的算法的运行时间是 $\mathcal{哦}(n^3\mathrm{日志}n)$，我们提供了一个下界 $\mathrm{\Omega }(n^2)$任何算法计算前缀Fréchet可相似的轮廓的运行时间，其中Ñ表示上段的数量Ť和${吨}^{\prime }$. 这意味着我们的运行时间至多是远离最优的一个接近线性的因素。