This work is motivated by several basic problems and techniques that rely on space decomposition of arrangements of hyperplanes in high-dimensional spaces, most notably Meiser’s 1993 algorithm (Meiser in Inf Comput 106(2):286–303, 1993) for point location in such arrangements. A standard approach to these problems is via random sampling, in which one draws a random sample of the hyperplanes, constructs a suitable decomposition of its arrangement, and recurses within each cell of the decomposition with the subset of hyperplanes that cross the cell. The efficiency of the resulting algorithm depends on the quality of the sample, which is controlled by various parameters. One of these parameters is the classical VC-dimension , and its associated primal shatter dimension , of a suitably defined corresponding range space. Another parameter, which we refer to here as the combinatorial dimension , is the maximum number of hyperplanes that are needed to define a cell that can arise in the decomposition of some sample of the input hyperplanes; this parameter arises in Clarkson’s (and later Clarkson and Shor’s) random sampling technique. We re-examine these parameters for the two main space decomposition techniques— bottom-vertex triangulation , and vertical decomposition , including their explicit dependence on the dimension d , and discover several unexpected phenomena, which show that, in both techniques, there are large gaps between the VC-dimension (and primal shatter dimension), and the combinatorial dimension. Our main application is to point location in an arrangement of n hyperplanes is $$\mathbb {R}^d$$ R d , in which we show that the query cost in Meiser’s algorithm can be improved if one uses vertical decomposition instead of bottom-vertex triangulation, at the cost of some increase in the preprocessing cost and storage (which seem to be stated incorrectly, and are not worked out, in Meiser’s work). Our improved bounds rely on establishing several new structural properties and improved complexity bounds for vertical decomposition, which are of independent interest, and which we expect to find additional applications.

中文翻译：

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超平面的分解排列：VC维、组合维和点位置

这项工作受到几个基本问​​题和技术的推动，这些问题和技术依赖于高维空间中超平面排列的空间分解，最着名的是 Meiser 1993 年的算法（Meiser in Inf Comput 106(2):286–303, 1993）在这样的安排。解决这些问题的标准方法是通过随机抽样，其中抽取超平面的随机样本，构建其排列的合适分解，并在分解的每个单元内递归使用跨单元的超平面子集。所得算法的效率取决于样本的质量，该质量受各种参数控制。这些参数之一是适当定义的相应范围空间的经典 VC 维度 及其相关的原始粉碎维度 。另一个参数，我们在这里称为组合维数，是定义一个单元所需的最大超平面数量，该单元可能出现在输入超平面的某些样本的分解中；这个参数出现在克拉克森（以及后来的克拉克森和肖尔）的随机抽样技术中。我们重新检查了两种主要空间分解技术（底顶点三角剖分和垂直分解）的这些参数，包括它们对维度 d 的显式依赖性，并发现了一些意想不到的现象，这表明这两种技术存在很大差距在 VC 维度（和原始粉碎维度）和组合维度之间。我们的主要应用是在 n 个超平面的排列中点位置为 $$\mathbb {R}^d$$ R d ，其中我们表明，如果使用垂直分解而不是底部顶点三角剖分，Meiser 算法中的查询成本可以得到改善，但代价是预处理成本和存储量有所增加（这似乎是错误的，并且不起作用）出，在迈泽的工作中）。我们改进的边界依赖于为垂直分解建立几个新的结构特性和改进的复杂性边界，它们是独立的，我们希望找到更多的应用。