$\renewcommand{\Re}{\mathbb{R}}\newcommand{\eps}{\varepsilon}\newcommand{\poly}{\mathrm{poly}} $In this paper, we study the problem of $L_1$-fitting a shape to a set of $n$ points in $\Re^d$ (where $d$ is a fixed constant), where the target is to minimize the sum of distances of the points to the shape, or the sum of squared distances. We present a general technique for computing a $(1 + \eps ) $-approximation for such a problem, with running time $O(n + \poly( \log n, 1/\eps))$, where $\poly(\log n, 1/\eps)$ is a polynomial of constant degree of $\log n$ and $1/\eps$ (the power of the polynomial is a function of $d$). The new algorithm runs in linear time for a fixed $\eps>0$, and is the first subquadratic algorithm for this problem. Applications of the algorithm include best fitting either a circle, a sphere, or a cylinder to a set of points when minimizing the sum of distances (or squared distances) to the respective shape.

翻译：$\renewcommand{\Re}{\mathbb{R}}\newcommand{\eps}{\varepsilon}\newcommand{\poly}{\mathrm{poly}}$ 本文研究了在$\Re^d$（其中$d$为固定常数）中，对一组$n$个点进行$L_1$形状拟合的问题，其目标是最小化各点到该形状的距离之和或距离平方之和。我们提出了一种通用技术，用于计算此类问题的$(1 + \eps)$近似解，其运行时间为$O(n + \poly( \log n, 1/\eps))$，其中$\poly(\log n, 1/\eps)$是$\log n$与$1/\eps$的常数阶多项式（多项式的次数是$d$的函数）。该新算法在固定$\eps>0$时具有线性运行时间，是首个针对此问题的亚二次算法。该算法的应用包括：在最小化到相应形状的距离（或距离平方）之和时，为点集最优拟合圆、球体或圆柱体。