Probabilistic and parallel algorithms for centroidal Voronoi tessellations with application to meshless computing and numerical analysis on surfaces

Centroidal Voronoi tessellations (CVT) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. Such tessellations are of use in very diverse applications, including data compression, clustering analysis, cell biology, territorial behavior of animals, optimal allocation of resources, and grid generation. A detailed review is given in chapter 1. In chapter 2, some probabilistic methods for determining centroidal Voronoi tessellations and their parallel implementation on distributed memory systems are presented. The results of computational experiments performed on a CRAY T3E-600 system are given for each algorithm. These demonstrate the superior sequential and parallel performance of a new algorithm we introduce. Then, new algorithms are presented in chapter 3 for the determination of point sets and associated support regions that can then be used in meshless computing methods. The algorithms are probabilistic in nature so that they are totally meshfree, i.e., they do not require, at any stage, the use of any coarse or fine boundary conforming or superimposed meshes. Computational examples are provided that show, for both uniform and non-uniform point distributions that the algorithms result in high-quality point sets and high-quality support regions. The extensions of centroidal Voronoi tessellations to general spaces and sets are also available. For example, tessellations of surfaces in a Euclidean space may be considered. In chapter 4, a precise definition of such constrained centroidal Voronoi tessellations (CCVT\u27s) is given and a number of their properties are derived, including their characterization as minimizers of a kind of energy. Deterministic and probabilistic algorithms for the construction of CCVT\u27s are presented and some analytical results for one of the algorithms are given. Some computational examples are provided which serve to illustrate the high quality of CCVT point sets. CCVT point sets are also applied to polynomial interpolation and numerical integration on the sphere. Finally, some conclusions are given in chapter 5

On Variants of k-means Clustering

\textit{Clustering problems} often arise in the fields like data mining, machine learning etc. to group a collection of objects into similar groups with respect to a similarity (or dissimilarity) measure. Among the clustering problems, specifically \textit{$k$-means} clustering has got much attention from the researchers. Despite the fact that $k$-means is a very well studied problem its status in the plane is still an open problem. In particular, it is unknown whether it admits a PTAS in the plane. The best known approximation bound in polynomial time is 9+\eps. In this paper, we consider the following variant of $k$-means. Given a set $C$ of points in $\mathcal{R}^d$ and a real $f > 0$, find a finite set $F$ of points in $\mathcal{R}^d$ that minimizes the quantity $f*|F|+\sum_{p\in C} \min_{q \in F} {||p-q||}^2$. For any fixed dimension $d$, we design a local search PTAS for this problem. We also give a "bi-criterion" local search algorithm for $k$-means which uses (1+\eps)k centers and yields a solution whose cost is at most (1+\eps) times the cost of an optimal $k$-means solution. The algorithm runs in polynomial time for any fixed dimension. The contribution of this paper is two fold. On the one hand, we are being able to handle the square of distances in an elegant manner, which yields near optimal approximation bound. This leads us towards a better understanding of the $k$-means problem. On the other hand, our analysis of local search might also be useful for other geometric problems. This is important considering that very little is known about the local search method for geometric approximation.Comment: 15 page

Zenith: Utility-Aware Resource Allocation for Edge Computing

In the Internet of Things(IoT) era, the demands for low-latency computing for time-sensitive applications (e.g., location-based augmented reality games, real-time smart grid management, real-time navigation using wearables) has been growing rapidly. Edge Computing provides an additional layer of infrastructure to fill latency gaps between the IoT devices and the back-end computing infrastructure. In the edge computing model, small-scale micro-datacenters that represent ad-hoc and distributed collection of computing infrastructure pose new challenges in terms of management and effective resource sharing to achieve a globally efficient resource allocation. In this paper, we propose Zenith, a novel model for allocating computing resources in an edge computing platform that allows service providers to establish resource sharing contracts with edge infrastructure providers apriori. Based on the established contracts, service providers employ a latency-aware scheduling and resource provisioning algorithm that enables tasks to complete and meet their latency requirements. The proposed techniques are evaluated through extensive experiments that demonstrate the effectiveness, scalability and performance efficiency of the proposed model

A simple linear time ( 1+ ε)- approximation algorithm for geometric k-means clustering in any dimensions

We present the first linear time (1+ε)-approximation algorithm for the k-means problem for fixed k and ε. Our algorithm runs in O(nd) time, which is linear in the size of the input. Another feature of our algorithm is its simplicity - the only technique involved is random sampling

How Fast is the k-means Method

We present polynomial upper and lower bounds on the number of iterations performed by the k-means method (a.k.a. Lloyd’s method) for k-means clustering. Our upper bounds are polynomial in the number of points, number of clusters, and the spread of the point set. We also present a lower bound, showing that in the worst case the k-means heuristic needs to perform Ω(n) iterations, for n points on the real line and two centers. Surprisingly, the spread of the point set in this construction is polynomial. This is the first construction showing that the k-means heuristic requires more than a polylogarithmic number of iterations. Furthermore, we present two alternative algorithms, with guaranteed performance, which are simple variants of the k-means method. Results of our experimental studies on these algorithms are also presented.