Planar β\beta-skeletons via point location in monotone subdivisions of subset of lunes

We present a new algorithm for lune-based $\beta$-skeletons for sets of $n$ points in the plane, for $\beta \in (2,\infty]$, the only case when optimal algorithms are not known. The running time of the algorithm is $O(n^{3/2} \log^{1/2} n)$, which is the best known and is an improvement of Rao and Mukhopadhyay \cite{rm97} result. The method is based on point location in monotonic subdivisions of arrangements of curve segments

Algorithmic and geometric aspects of data depth with focus on β\beta-skeleton depth

The statistical rank tests play important roles in univariate non-parametric data analysis. If one attempts to generalize the rank tests to a multivariate case, the problem of defining a multivariate order will occur. It is not clear how to define a multivariate order or statistical rank in a meaningful way. One approach to overcome this problem is to use the notion of data depth which measures the centrality of a point with respect to a given data set. In other words, a data depth can be applied to indicate how deep a point is located with respect to a given data set. Using data depth, a multivariate order can be defined by ordering the data points according to their depth values. Various notions of data depth have been introduced over the last decades. In this thesis, we discuss three depth functions: two well-known depth functions halfspace depth and simplicial depth, and one recently defined depth function named as $\beta$-skeleton depth, $\beta\geq 1$. The $\beta$-skeleton depth is equivalent to the previously defined spherical depth and lens depth when $\beta=1$ and $\beta=2$, respectively. Our main focus in this thesis is to explore the geometric and algorithmic aspects of $\beta$-skeleton depth.Comment: PhD thesis By Rasoul Shahsavarifar at the Faculty of Computer Science, UN