相对$(p,ε)$ -几何近似
Relative $(p,ε)$-Approximations in Geometry
摘要：
我们重新检查的概念相对$(p,\eps)$ -近似，最近在[CKMS 06]，并建立其大小的上限，在一般的范围空间有限VC维，使用抽样理论[LLS 01]和在几个早期的研究[Pol 86，Hau 92，Tal 94]。我们还调查了不同的采样概念，用于计算几何，学习和其他领域，并显示它们是如何相互关联的。然后，我们给出了较小尺寸的相对$(p,\eps)$ -近似的范围空间，涉及点和半空间在两个和更高的维度的建设。平面构造基于一种新的结构--相对交叉数较小的生成树，我们认为这是独立感兴趣的。相对$(p,\eps)$ -近似出现在几个几何问题，如近似范围计数，我们应用我们的新结构，以获得有效的解决方案，在三维近似范围计数。我们还提出了一个简单的解决方案的平面的情况下。
英文摘要：
We re-examine the notion of relative $(p,\eps)$-approximations, recently introduced in [CKMS06], and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in [LLS01] and in several earlier studies [Pol86, Hau92, Tal94]. We also survey the different notions of sampling, used in computational geometry, learning, and other areas, and show how they relate to each other. We then give constructions of smaller-size relative $(p,\eps)$-approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure--spanning trees with small relative crossing number, which we believe to be of independent interest. Relative $(p,\eps)$-approximations arise in several geometric problems, such as approximate range counting, and we apply our new structures to obtain efficient solutions for approximate range counting in three dimensions. We also present a simple solution for the planar case.
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