Measuring the similarity of two polygonal curves is a fundamental computational task. Among alternatives, the Frechet distance is one of the most well-studied similarity measures. Informally, the Frechet distance is described as the minimum leash length required for a man on one of the curves to walk a dog on the other curve continuously from the starting to the ending points. In this paper we study a variant called the Frechet gap distance. In the man and dog analogy, the Frechet gap distance minimizes the difference of the longest and smallest leash lengths used over the entire walk. This measure in some ways better captures our intuitive notions of curve similarity, for example giving distance zero to translated copies of the same curve. The Frechet gap distance was originally introduced by Filtser and Katz [19] in the context of the discrete Frechet distance. Here we study the continuous version, which presents a number of additional challenges not present in discrete case. In particular, the continuous nature makes bounding and searching over the critical events a rather difficult task. For this problem we give an $$O(n^5\log n)$$ time exact algorithm and a more efficient $$O(n^2\log n +({n^2}/{{\varepsilon }})\log ({1}/{{\varepsilon }}))$$ time $$(1+{\varepsilon })$$ -approximation algorithm, where n is the total number of vertices of the input curves. Note that, ignoring logarithmic factors, for any constant $${\varepsilon }$$ our approximation has quadratic running time, matching the lower bound, assuming SETH [Bringmann, K.: Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails (2014)], for approximating the standard Frechet distance for general curves.

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计算 Fréchet 间隙距离

测量两条多边形曲线的相似性是一项基本的计算任务。在备选方案中，Frechet 距离是研究最充分的相似性度量之一。非正式地，Frechet 距离被描述为一个人在一条曲线上从起点到终点连续在另一条曲线上遛狗所需的最小皮带长度。在本文中，我们研究了一种称为 Frechet 间隙距离的变体。在人和狗的类比中，Frechet 间隙距离最小化了整个步行过程中使用的最长和最小皮带长度的差异。这种度量在某些方面更好地捕捉了我们对曲线相似性的直观概念，例如将距离为零到同一曲线的平移副本。Frechet 间隙距离最初是由 Filtser 和 Katz [19] 在离散 Frechet 距离的背景下引入的。在这里，我们研究连续版本，它提出了一些在离散情况下不存在的额外挑战。特别是，连续性使得对关键事件进行边界和搜索成为一项相当困难的任务。对于这个问题，我们给出了一个 $$O(n^5\log n)$$ 时间精确算法和一个更有效的 $$O(n^2\log n +({n^2}/{{\varepsilon }} )\log ({1}/{{\varepsilon }}))$$ time $$(1+{\varepsilon })$$ - 近似算法，其中 n 是输入曲线的顶点总数。请注意，忽略对数因子，对于任何常数 $${\varepsilon }$$ 我们的近似值具有二次运行时间，匹配下界，假设 SETH [Bringmann, K.：为什么遛狗需要时间：