多边形曲线的近似$(k,\ell)$ -中值聚类
Approximating $(k,\ell)$-Median Clustering for Polygonal Curves
摘要：
2015年，Driemel、Krivošija和Sohler引入了$(k,\ell)$ -中值问题，用于在Fréchet距离下聚类多边形曲线。给定一组输入曲线，该问题要求找到至多$\ell$顶点的$k$中值曲线，每个顶点最小化所有输入曲线上到其最近中值曲线的Fréchet距离之和。其算法的一个主要缺点是输入曲线被限制在实直线上。在本文中，我们提出了一种随机双准则近似算法，适用于多边形曲线在$\mathbb{R}^d$和实现近似因子$(1+ε)$的聚类成本。该算法在曲线数量方面具有最坏情况的运行时间线性，在每条曲线的最大顶点数量方面具有多项式，即它们的复杂性，并且在$d$、$\ell$、$ε$和$δ$中具有指数，即，失败的概率。我们通过一个捷径引理来实现这一结果，该引理保证了多边形曲线的存在，该多边形曲线具有与复杂性$\ell$的最佳中值曲线相似的成本，但复杂性至多为$2\ell-2$，并且其顶点可以有效地计算。我们将这个引理与Kumar等人的超集采样技术相结合，得出我们的聚类结果。在此过程中，我们描述和分析了Ackermann等人的算法的推广，这可能是独立的利益。
英文摘要：
In 2015, Driemel, Krivošija and Sohler introduced the $(k,\ell)$-median problem for clustering polygonal curves under the Fréchet distance. Given a set of input curves, the problem asks to find $k$ median curves of at most $\ell$ vertices each that minimize the sum of Fréchet distances over all input curves to their closest median curve. A major shortcoming of their algorithm is that the input curves are restricted to lie on the real line. In this paper, we present a randomized bicriteria-approximation algorithm that works for polygonal curves in $\mathbb{R}^d$ and achieves approximation factor $(1+ε)$ with respect to the clustering costs. The algorithm has worst-case running-time linear in the number of curves, polynomial in the maximum number of vertices per curve, i.e. their complexity, and exponential in $d$, $\ell$, $ε$ and $δ$, i.e., the failure probability. We achieve this result through a shortcutting lemma, which guarantees the existence of a polygonal curve with similar cost as an optimal median curve of complexity $\ell$, but of complexity at most $2\ell-2$, and whose vertices can be computed efficiently. We combine this lemma with the superset-sampling technique by Kumar et al. to derive our clustering result. In doing so, we describe and analyze a generalization of the algorithm by Ackermann et al., which may be of independent interest.
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