An algorithm to compute CVTs for finitely generated Cantor distributions

Abstract

Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions with respect to a given probability measure. CVT is a fundamental notion that has a wide spectrum of applications in computational science and engineering. In this paper, an algorithm is given to obtain the CVTs with $n$-generators to level $m$, for any positive integers $m$ and $n$, of any Cantor set generated by a pair of self-similar mappings given by $S_1(x)=r_{1}x$ and $S_2(x)=r_{2}x+(1-r_2)$ for $x\in \mathbb{R}$, where $r_1,r_2>0$ and $r_1+r_2<1$, with respect to any probability distribution $P$ such that $P=p_{1}P\circ S_1^{-1}+p_{2}P\circ S_2^{-1}$, where $p_1,p_2>0$ and $p_1+p_2=1$.