Uniform distribution of the points has been of interest to researchers for a long time and has applications in different areas of Mathematics and Computer Science. One of the well-known measures to evaluate the uniformity of a given distribution is Discrepancy, which assesses the difference between the Uniform distribution and the empirical distribution given by putting mass points at the points of the given set. While Discrepancy is very useful to measure uniformity, it is computationally challenging to be calculated accurately. We introduce the concept of directed Discrepancy based on which we have developed an algorithm, called Directional Discrepancy, that can offer accurate approximation for the cap Discrepancy of a finite set distributed on the unit Sphere, $\mathbb{S}^2.$ We also analyze the time complexity of the Directional Discrepancy algorithm precisely; and practically evaluate its capacity by calculating the Cap Discrepancy of a specific distribution, Polar Coordinates, which aims to distribute points uniformly on the Sphere.

中文翻译：

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一种计算上限差异的实用算法

长期以来，研究人员一直对点的均匀分布感兴趣，并且在数学和计算机科学的不同领域都有应用。评估给定分布均匀性的众所周知的度量之一是差异，它评估均匀分布与通过将质量点放在给定集合的点上给出的经验分布之间的差异。虽然差异对于测量均匀性非常有用，但准确计算在计算上具有挑战性。我们引入了定向差异的概念，在此基础上我们开发了一种称为定向差异的算法，该算法可以为分布在单位 Sphere $\mathbb{S}^2 上的有限集的上限差异提供准确的近似值。$ 我们还精确地分析了 Directional Discrepancy 算法的时间复杂度；并通过计算特定分布极坐标的上限差异来实际评估其容量，该分布旨在在球体上均匀分布点。