c-填充曲线上Fréchet距离的改进逼近
Improved approximation for Fréchet distance on c-packed curves matching conditional lower bounds
摘要：
Fréchet距离是一个研究得很好的，非常流行的两条曲线相似性的度量。最著名的算法具有二次时间复杂度，最近已被证明是最佳假设强指数时间假设（SETH）[Bringmann FOCS'14]。为了克服最坏情况下的二次时间障碍，已经研究了限制类的曲线，试图捕捉现实的输入曲线。最流行的这类曲线是c-填充曲线，Fréchet距离在时间上具有$(1+ε)$ -近似值$\tilde{O}(c n /ε)$ [Driemel et al. DCG'12]。在$d \ge 5$维度中，对于任何$δ> 0$，这不能改进为$O((cn/\sqrtε)^{1-δ})$，除非SETH失败[Bringmann FOCS'14]。在本文中，利用属性，防止更强的下限，我们提出了一个改进的算法与运行时$\tilde{O}(cn/\sqrtε)$。这是最佳的高维度，除了低阶因素，除非SETH失败。我们的主要新成分如下：为了填充经典的自由空间图，我们将短的子曲线投影到一条线上，这会产生一维分离曲线，顶点之间的成对距离大致相同。然后，我们处理这种特殊情况下，在近线性时间仔细扩展的贪心算法的Fréchet距离的一维分离曲线。
英文摘要：
The Fréchet distance is a well-studied and very popular measure of similarity of two curves. The best known algorithms have quadratic time complexity, which has recently been shown to be optimal assuming the Strong Exponential Time Hypothesis (SETH) [Bringmann FOCS'14]. To overcome the worst-case quadratic time barrier, restricted classes of curves have been studied that attempt to capture realistic input curves. The most popular such class are c-packed curves, for which the Fréchet distance has a $(1+ε)$-approximation in time $\tilde{O}(c n /ε)$ [Driemel et al. DCG'12]. In dimension $d \ge 5$ this cannot be improved to $O((cn/\sqrtε)^{1-δ})$ for any $δ> 0$ unless SETH fails [Bringmann FOCS'14]. In this paper, exploiting properties that prevent stronger lower bounds, we present an improved algorithm with runtime $\tilde{O}(cn/\sqrtε)$. This is optimal in high dimensions apart from lower order factors unless SETH fails. Our main new ingredients are as follows: For filling the classical free-space diagram we project short subcurves onto a line, which yields one-dimensional separated curves with roughly the same pairwise distances between vertices. Then we tackle this special case in near-linear time by carefully extending a greedy algorithm for the Fréchet distance of one-dimensional separated curves.
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