Given n data points in ${\mathbb{R}}^d$, an appropriate edge-weighted graph connecting the data points finds application in solving clustering, classification, and regression problems. The graph proposed by Daitch, Kelner and Spielman (ICML 2009) can be computed by quadratic programming and hence in polynomial time. While a more efficient algorithm would be preferable, replacing quadratic programming is challenging even for the special case of points in one dimension. We develop a dynamic programming algorithm for this case that runs in $O(n^2)$ time under the Real-RAM model, where arithmetic on real numbers takes constant time.

中文翻译：

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将图拟合到一维数据

给定n个数据点${\mathbb{[R}}^d$，将数据点连接起来的合适的边缘加权图可以解决聚类，分类和回归问题。Daitch，Kelner和Spielman（ICML 2009）提出的图形可以通过二次编程来计算，因此可以在多项式时间内进行计算。尽管更有效的算法是可取的，但即使对于一维点的特殊情况，替换二次规划也是一个挑战。我们针对这种情况开发了一种动态编程算法，该算法可在$Ø（{ñ}^{\mathrm{2个}}）$ 在Real-RAM模型下的时间，其中实数的运算需要固定的时间。