在$d$维中近似离散和连续中线段
Approximating the discrete and continuous median line segments in $d$ dimensions
摘要：
考虑$\mathbb{R}^d$中$n$点的集合$P$。在离散中值线段问题中，目标是找到由$P$中的一对点限定的线段，使得从$P$到该线段的欧几里德距离之和最小。在连续中值线段问题中，给定一个实数$\ell>0$，目标是在$\mathbb{R}^d$中定位一个长度为$\ell$的线段，使得$P$与该线段之间的欧氏距离之和最小.我们将展示如何计算$(1+εΔ)$ -和$(1+ε)$ -近似的离散中位线段的时间分别为$O(nε^{-2d}\log n)$和$O(n^2ε^{-d})$，其中$Δ$是由点对跨越的线段的传播。在开发我们的算法时，通过使用对分解的原理，我们推导出新的数据结构，使我们能够快速近似从一组点到给定线段或点的距离之和。据我们所知，我们利用对分解解决最小和设施选址问题是第一次，它是通用的，易于实现。证明了对于平面上的$n\geq3$非共线点，仅用直尺和圆规是不可能构造出连续的中线线段的。鉴于此，我们提出了一个$O(n^dε^{-d})$时间算法，用于在$1+ε$的一个因子内逼近$\mathbb{R}^d$中的连续中线段。该算法是基于推广的点段对分解从离散到连续域。最后，我们给出了一个$(1+ε)$ -近似算法，其时间复杂度是次二次的$n$，用于求解$\mathbb{R}^2$中的约束中线段问题，其中中线段的端点或斜率在输入处被给定。
英文摘要：
Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line segment is minimized. In the continuous median line segment problem, a real number $\ell>0$ is given, and the goal is to locate a line segment of length $\ell$ in $\mathbb{R}^d$ such that the sum of the Euclidean distances between $P$ and the line segment is minimized. We show how to compute $(1+εΔ)$- and $(1+ε)$-approximations to a discrete median line segment in time $O(nε^{-2d}\log n)$ and $O(n^2ε^{-d})$, respectively, where $Δ$ is the spread of line segments spanned by pairs of points. While developing our algorithms, by using the principle of pair decomposition, we derive new data structures that allow us to quickly approximate the sum of the distances from a set of points to a given line segment or point. To our knowledge, our utilization of pair decompositions for solving minsum facility location problems is the first of its kind; it is versatile and easily implementable. We prove that it is impossible to construct a continuous median line segment for $n\geq3$ non-collinear points in the plane by using only ruler and compass. In view of this, we present an $O(n^dε^{-d})$-time algorithm for approximating a continuous median line segment in $\mathbb{R}^d$ within a factor of $1+ε$. The algorithm is based upon generalizing the point-segment pair decomposition from the discrete to the continuous domain. Last but not least, we give an $(1+ε)$-approximation algorithm, whose time complexity is sub-quadratic in $n$, for solving the constrained median line segment problem in $\mathbb{R}^2$ where an endpoint or the slope of the median line segment is given at input.
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