The Geometry of Scheduling

We consider the following general scheduling problem: The input consists of n jobs, each with an arbitrary release time, size, and a monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as weighted flow, weighted tardiness, and sum of flow squared. Our main result is a randomized polynomial-time algorithm with an approximation ratio O(log log nP), where P is the maximum job size. We also give an O(1) approximation in the special case when all jobs have identical release times. The main idea is to reduce this scheduling problem to a particular geometric set-cover problem which is then solved using the local ratio technique and Varadarajan's quasi-uniform sampling technique. This general algorithmic approach improves the best known approximation ratios by at least an exponential factor (and much more in some cases) for essentially all of the nontrivial common special cases of this problem. Our geometric interpretation of scheduling may be of independent interest.Comment: Conference version in FOCS 201

Worst-case User Analysis in Poisson Voronoi Cells

In this letter, we focus on the performance of a worst-case mobile user (MU) in the downlink cellular network. We derive the coverage probability and the spectral efficiency of the worst-case MU using stochastic geometry. Through analytical and numerical results, we draw out interesting insights that the coverage probability and the spectral efficiency of the worst-case MU decrease down to 23% and 19% of those of a typical MU, respectively. By applying a coordinated scheduling (CS) scheme, we also investigate how much the performance of the worst-case MU is improved.Comment: Accepted, IEEE Communications Letter

PARALLEL COMPUTATION DESIGN OF DETERMINANT AND INVERSE OF A MATRICE USING COMPUTATIONAL GEOMETRY ANALYSIS TECHNIQUE

The aim of this research is to study the use of computational geometry analysis technique in detail to design a parallel computation to determine determinant and inverse of a matrice using Gauss-Jordan method. In comparing with dependence graph technique, it is well known that computational geometry analysis gives us an  easier way in determining determinant and inverse of a matrice, moreover for handling a computation domain with dimension of higher than 3. The type of scheduling function used in this research is of linear type. Some important facts are discovered in this research, the most important one is, this technique gives us the ease in arranging data scheduling scheme based on the observation of the computation domain. The other important fact is, it is also give us the ease to understand the scheme of the data scheduling since this technique can be collaborated with dependence graph technique for visualizing the data schedulin