度量空间中点样本均匀性的间隔比方法
Uniformity of point samples in metric spaces using gap ratio
摘要：
Teramoto等人定义了一种称为间隙比的新度量，该度量度量从$\cal S$（$\mathbb{R}^2$的有界子集）采样的有限点集的均匀性。我们通过利用覆盖和填充半径将测度的定义推广到所有度量空间。间隙比的定义只需要一个度量，而不像差异，一个广泛使用的均匀性度量，它取决于范围空间及其体积的概念。我们还显示了一些有趣的连接间隙比德洛内三角剖分和欧几里德平面的差异。这项工作的主要重点是解决优化相关的问题，从度量空间中选择均匀的点样本，使用间隙比测量的均匀性。我们考虑离散空间，如图和点集在欧几里德空间和连续空间，如单位平方和道路连通空间。我们推导出下界，证明硬度和近似硬度的结果。我们表明，一个一般的近似算法框架，给出不同的度量空间的基础上，我们推导出的下界不同的近似比。除了上述，我们证明存在的coresets采样均匀点从欧氏空间-静态和流的情况下。这导致了一个$\left( 1+ε\right)$ -近似算法的均匀采样从欧几里德空间。
英文摘要：
Teramoto et al. defined a new measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a bounded subset of $\mathbb{R}^2$. We generalize this definition of measure over all metric spaces by appealing to covering and packing radius. The definition of gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion of a range space and its volume. We also show some interesting connections of gap ratio to Delaunay triangulation and discrepancy in the Euclidean plane. The major focus of this work is on solving optimization related questions about selecting uniform point samples from metric spaces; the uniformity being measured using gap ratio. We consider discrete spaces like graph and set of points in the Euclidean space and continuous spaces like the unit square and path connected spaces. We deduce lower bounds, prove hardness and approximation hardness results. We show that a general approximation algorithm framework gives different approximation ratios for different metric spaces based on the lower bound we deduce. Apart from the above, we show existence of coresets for sampling uniform points from the Euclidean space -- for both the static and the streaming case. This leads to a $\left( 1+ε\right)$-approximation algorithm for uniform sampling from the Euclidean space.
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