曲线的近似$(k,\ell)$ -中心聚类
Approximating $(k,\ell)$-center clustering for curves
摘要：
欧几里德$k$中心问题是计算机科学中一个被广泛研究的经典问题.给定欧几里德空间中的$n$点的集合$\mathcal{G}$，问题是确定$k$中心（不一定是$\mathcal{G}$的一部分）的集合$\mathcal{C}$，使得$\mathcal{G}$中的点与其在$\mathcal{C}$中的最近邻点之间的最大距离最小化。本文研究了Fréchet距离下多边形曲线的$(k,\ell)$ -中心问题，即给定$\mathbb{R}^d$中的$n$多边形曲线的集合$\mathcal{G}$，每个$m$，确定$\mathbb{R}^d$中的$k$多边形曲线的集合$\mathcal{C}$，每个$\ell$，使得$\mathcal{G}$中的曲线到$\mathcal{C}$中最近曲线的最大Fréchet距离最小.在本文中，我们大大扩展和改进已知的逼近边界的曲线在$2$和更高的维度。我们表明，如果$\ell$是输入的一部分，则没有多项式时间近似方案，除非$\mathsf{P}=\mathsf{NP}$。我们的结构产生不同的边界的一维和二维曲线和离散和连续的Fréchet距离。在二维曲线上的离散Fréchet距离的情况下，我们显示出接近$2.598$的因子内的近似的硬度。当$k=1$和$\mathsf{NP}$硬度扩展到$\ell=\infty$，即，计算Fréchet距离下的最小包围球的问题。最后，我们观察到，冈萨雷斯的算法与曲线简化相结合的一个仔细的适应产生$3$ -近似在任何维度，提供了一个最佳的简化可以精确计算。我们的结论是，我们的近似边界接近紧。
英文摘要：
The Euclidean $k$-center problem is a classical problem that has been extensively studied in computer science. Given a set $\mathcal{G}$ of $n$ points in Euclidean space, the problem is to determine a set $\mathcal{C}$ of $k$ centers (not necessarily part of $\mathcal{G}$) such that the maximum distance between a point in $\mathcal{G}$ and its nearest neighbor in $\mathcal{C}$ is minimized. In this paper we study the corresponding $(k,\ell)$-center problem for polygonal curves under the Fréchet distance, that is, given a set $\mathcal{G}$ of $n$ polygonal curves in $\mathbb{R}^d$, each of complexity $m$, determine a set $\mathcal{C}$ of $k$ polygonal curves in $\mathbb{R}^d$, each of complexity $\ell$, such that the maximum Fréchet distance of a curve in $\mathcal{G}$ to its closest curve in $\mathcal{C}$ is minimized. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension $2$ and higher. We show that, if $\ell$ is part of the input, then there is no polynomial-time approximation scheme unless $\mathsf{P}=\mathsf{NP}$. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fréchet distance. In the case of the discrete Fréchet distance on two-dimensional curves, we show hardness of approximation within a factor close to $2.598$. This result also holds when $k=1$, and the $\mathsf{NP}$-hardness extends to the case that $\ell=\infty$, i.e., for the problem of computing the minimum-enclosing ball under the Fréchet distance. Finally, we observe that a careful adaptation of Gonzalez' algorithm in combination with a curve simplification yields a $3$-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.
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