关于计算$k$ -$k$ -Fréchet距离
On Computing the $k$-Shortcut Fréchet Distance
摘要：
Fréchet距离是多边形曲线相异性的常用度量。它被定义为一个最小-最大公式，考虑所有的方向保持连续双射的两条曲线。由于其对噪声的敏感性，Driemel和Har-Peled在2012年引入了快捷Fréchet距离，其中允许沿着其中一条曲线采取快捷方式，类似于序列的编辑距离。我们分析了这个问题的参数化版本，其中的捷径的数量是由一个参数$k$。相应的判定问题可以表述如下：给定至多$n$顶点的两条多边形曲线$T$和$B$，参数$k$和距离阈值$δ$，是否可以沿着$B$引入$k$捷径，使得所得曲线和曲线$T$的Fréchet距离至多为$δ$？我们研究这个问题的多边形曲线在平面上。我们提供了一个复杂性分析，这个问题有以下结果：（i）假设指数时间假设（ETH），不存在算法的运行时间由$n^{o(k)}$限制;（ii）存在一个决策算法的运行时间在$O(kn^{2k+2}\log n)$。相反，我们还表明，有效的近似决策算法是可能的，即使当$k$是大的。对于固定的$\varepsilon$，我们在$O(k n^2 \log^2 n)$中提出了一个具有运行时间的$(3+\varepsilon)$近似判决器算法.此外，我们还可以证明，如果$k$是一个常数，并且对于某个常数$c$，两条曲线是$c$填充的，则近似决策器算法在近似线性时间内运行。
英文摘要：
The Fréchet distance is a popular measure of dissimilarity for polygonal curves. It is defined as a min-max formulation that considers all direction-preserving continuous bijections of the two curves. Because of its susceptibility to noise, Driemel and Har-Peled introduced the shortcut Fréchet distance in 2012, where one is allowed to take shortcuts along one of the curves, similar to the edit distance for sequences. We analyse the parameterized version of this problem, where the number of shortcuts is bounded by a parameter $k$. The corresponding decision problem can be stated as follows: Given two polygonal curves $T$ and $B$ of at most $n$ vertices, a parameter $k$ and a distance threshold $δ$, is it possible to introduce $k$ shortcuts along $B$ such that the Fréchet distance of the resulting curve and the curve $T$ is at most $δ$? We study this problem for polygonal curves in the plane. We provide a complexity analysis for this problem with the following results: (i) assuming the exponential-time-hypothesis (ETH), there exists no algorithm with running time bounded by $n^{o(k)}$; (ii) there exists a decision algorithm with running time in $O(kn^{2k+2}\log n)$. In contrast, we also show that efficient approximate decider algorithms are possible, even when $k$ is large. We present a $(3+\varepsilon)$-approximate decider algorithm with running time in $O(k n^2 \log^2 n)$ for fixed $\varepsilon$. In addition, we can show that, if $k$ is a constant and the two curves are $c$-packed for some constant $c$, then the approximate decider algorithm runs in near-linear time.
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