We provide a general framework for getting expected linear time constant factor approximations (and in many cases FPTASs) to several well-known problems in Computational Geometry, such as $k$-center clustering and farthest nearest neighbor. The new approach is robust to variations in the input problem, and yet it is simple, elegant, and practical. In particular, many of these well-studied problems, which fit easily into our framework, either previously had no linear time approximation algorithm, or required rather involved algorithms and analysis. A short list of the problems we consider includes farthest nearest neighbor, $k$-center clustering, smallest disk enclosing $k$ points, Hausdorff distance, $k$th largest distance, $k$th smallest $m$-nearest neighbor distance, $k$th heaviest edge in the MST, and other spanning-forest type problems, problems involving upward closed set systems, and more. Finally, we show how to extend our framework such that the linear running time bound holds with high probability.

翻译：我们提出了一个通用框架，用于为计算几何中的若干经典问题（如$k$-中心聚类和最远最近邻问题）获得期望线性时间的常数因子近似解（在许多情况下甚至是完全多项式时间近似方案）。这一新方法对输入问题的变化具有鲁棒性，同时兼具简洁性、优雅性和实用性。特别地，许多已深入研究且易于纳入本框架的问题，此前要么缺乏线性时间近似算法，要么需要极为复杂的算法设计与分析。我们所处理的问题包括但不限于：最远最近邻、$k$-中心聚类、包含$k$点的最小圆盘、豪斯多夫距离、第$k$大距离、第$k$小$m$-最近邻距离、最小生成树中第$k$重边及其他生成森林类问题、涉及上闭集系统的问题等。最后，我们展示了如何扩展本框架，使得线性时间复杂度界能以高概率成立。