A practical algorithm to calculate Cap Discrepancy
Abstract
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 wellknown 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.10454
 arXiv:
 arXiv:2010.10454
 Bibcode:
 2020arXiv201010454B
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Data Structures and Algorithms;
 Computer Science  Machine Learning;
 Mathematics  Numerical Analysis;
 Mathematics  Optimization and Control
 EPrint:
 24 pages