Monte carlo methods for the rectilinear crossing number
Tutor / director / evaluatorHuemer, Clemens
Document typeBachelor thesis
Rights accessOpen Access
The thesis is dedicated to ﬁnd a fast Monte Carlo algorithm for the calculation of the rectilinear crossing number Cr(S) of a point set S in the plane, where Cr(S) is the number of intersections of all the straight line segments which connect pairs of points of the set. Crossing numbers are a central topic of research in the area of Discrete and Computational Geometry. A quadratic time algorithm to calculate Cr(S) is known, which, for large input size, is very time-consuming. We propose fast Monte Carlo algorithms to produce approximate solutions. To our knowledge, Monte Carlo methods have not been applied before in this setting. There is a trade-off between the precision of the approximated crossing number and the running time of the Monte Carlo algorithm. Since the outputs of Monte Carlo methods follow a normal distribution when the samples are independent, the exactitude of a Monte Carlo method is related with the variance. If the variance is smaller, the required sample size to reach a predefined precision of the solution (with high probability) will be smaller and the exactitude will be higher. In this thesis we introduce six Monte Carlo methods for the calculation of the rectilinear crossing number, and study their variances and the required sample sizes. Computational experiments conﬁrm the obtained theoretical results. Also, we apply some variance reduction techniques, such as importance sampling, antithetic variates, and control variates, to enhance the performance of the developed Monte Carlo methods. The best reduction technique in this thesis is the control variates technique which reduces the required sample size signiﬁcantly compared to the original method.