The discrete Fréchet distance is a popular measure for comparing polygonal curves. An important variant is the discrete Fréchet distance under translation, which enables detection of similar movement patterns in different spatial domains. For polygonal curves of length n in the plane, the fastest known algorithm runs in time Õ( n 5 ) [12]. This is achieved by constructing an arrangement of disks of size Õ( n 4 ), and then traversing its faces while updating reachability in a directed grid graph of size N := Õ( n 5 ), which can be done in time Õ(√ N ) per update [27]. The contribution of this article is two-fold. First, although it is an open problem to solve dynamic reachability in directed grid graphs faster than Õ(√ N ), we improve this part of the algorithm: We observe that an offline variant of dynamic s - t -reachability in directed grid graphs suffices, and we solve this variant in amortized time Õ( N 1/3 ) per update, resulting in an improved running time of Õ( N 4.66 ) for the discrete Fréchet distance under translation. Second, we provide evidence that constructing the arrangement of size Õ( N 4 ) is necessary in the worst case by proving a conditional lower bound of n 4 - o(1) on the running time for the discrete Fréchet distance under translation, assuming the Strong Exponential Time Hypothesis.

中文翻译：

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平移下的离散 Fréchet 距离

离散 Fréchet 距离是比较多边形曲线的常用度量。一个重要的变体是平移下的离散 Fréchet 距离，它可以检测不同空间域中的相似运动模式。对于长度的多边形曲线n在平面上，最快的已知算法及时运行 Õ(n 5) [12]。这是通过构造大小为 Õ(n 4)，然后遍历它的面，同时在大小的有向网格图中更新可达性ñ:= Õ(n 5), 可以及时完成 Õ(√ñ) 每次更新 [27]。这篇文章的贡献是双重的。首先，尽管在有向网格图中解决动态可达性比Õ(√ñ)，我们改进了这部分算法：我们观察到动态的离线变体s-吨- 有向网格图中的可达性就足够了，我们在摊销时间内解决了这个变体 Õ(ñ 1/3) 每次更新，从而提高 Õ(ñ 4.66) 用于平移下的离散 Fréchet 距离。其次，我们提供了构造尺寸 Õ(ñ 4) 在最坏的情况下是必要的，通过证明的条件下界n 4 - o(1)假设强指数时间假设，关于平移下离散 Fréchet 距离的运行时间。