37,098 research outputs found
Approximation algorithm for the kinetic robust K-center problem
AbstractTwo complications frequently arise in real-world applications, motion and the contamination of data by outliers. We consider a fundamental clustering problem, the k-center problem, within the context of these two issues. We are given a finite point set S of size n and an integer k. In the standard k-center problem, the objective is to compute a set of k center points to minimize the maximum distance from any point of S to its closest center, or equivalently, the smallest radius such that S can be covered by k disks of this radius. In the discrete k-center problem the disk centers are drawn from the points of S, and in the absolute k-center problem the disk centers are unrestricted.We generalize this problem in two ways. First, we assume that points are in continuous motion, and the objective is to maintain a solution over time. Second, we assume that some given robustness parameter 0<t⩽1 is given, and the objective is to compute the smallest radius such that there exist k disks of this radius that cover at least ⌈tn⌉ points of S. We present a kinetic data structure (in the KDS framework) that maintains a (3+ε)-approximation for the robust discrete k-center problem and a (4+ε)-approximation for the robust absolute k-center problem, both under the assumption that k is a constant. We also improve on a previous 8-approximation for the non-robust discrete kinetic k-center problem, for arbitrary k, and show that our data structure achieves a (4+ε)-approximation. All these results hold in any metric space of constant doubling dimension, which includes Euclidean space of constant dimension
Measuring Blood Glucose Concentrations in Photometric Glucometers Requiring Very Small Sample Volumes
Glucometers present an important self-monitoring tool for diabetes patients
and therefore must exhibit high accu- racy as well as good usability features.
Based on an invasive, photometric measurement principle that drastically
reduces the volume of the blood sample needed from the patient, we present a
framework that is capable of dealing with small blood samples, while
maintaining the required accuracy. The framework consists of two major parts:
1) image segmentation; and 2) convergence detection. Step 1) is based on
iterative mode-seeking methods to estimate the intensity value of the region of
interest. We present several variations of these methods and give theoretical
proofs of their convergence. Our approach is able to deal with changes in the
number and position of clusters without any prior knowledge. Furthermore, we
propose a method based on sparse approximation to decrease the computational
load, while maintaining accuracy. Step 2) is achieved by employing temporal
tracking and prediction, herewith decreasing the measurement time, and, thus,
improving usability. Our framework is validated on several real data sets with
different characteristics. We show that we are able to estimate the underlying
glucose concentration from much smaller blood samples than is currently
state-of-the- art with sufficient accuracy according to the most recent ISO
standards and reduce measurement time significantly compared to
state-of-the-art methods
Efficient iterative method for solving the Dirac-Kohn-Sham density functional theory
We present for the first time an efficient iterative method to directly solve
the four-component Dirac-Kohn-Sham (DKS) density functional theory. Due to the
existence of the negative energy continuum in the DKS operator, the existing
iterative techniques for solving the Kohn-Sham systems cannot be efficiently
applied to solve the DKS systems. The key component of our method is a novel
filtering step (F) which acts as a preconditioner in the framework of the
locally optimal block preconditioned conjugate gradient (LOBPCG) method. The
resulting method, dubbed the LOBPCG-F method, is able to compute the desired
eigenvalues and eigenvectors in the positive energy band without computing any
state in the negative energy band. The LOBPCG-F method introduces mild extra
cost compared to the standard LOBPCG method and can be easily implemented. We
demonstrate our method in the pseudopotential framework with a planewave basis
set which naturally satisfies the kinetic balance prescription. Numerical
results for Pt, Au, TlF, and BiSe indicate that the
LOBPCG-F method is a robust and efficient method for investigating the
relativistic effect in systems containing heavy elements.Comment: 31 pages, 5 figure
Topological Stability of Kinetic -Centers
We study the -center problem in a kinetic setting: given a set of
continuously moving points in the plane, determine a set of (moving)
disks that cover at every time step, such that the disks are as small as
possible at any point in time. Whereas the optimal solution over time may
exhibit discontinuous changes, many practical applications require the solution
to be stable: the disks must move smoothly over time. Existing results on this
problem require the disks to move with a bounded speed, but this model is very
hard to work with. Hence, the results are limited and offer little theoretical
insight. Instead, we study the topological stability of -centers.
Topological stability was recently introduced and simply requires the solution
to change continuously, but may do so arbitrarily fast. We prove upper and
lower bounds on the ratio between the radii of an optimal but unstable solution
and the radii of a topologically stable solution---the topological stability
ratio---considering various metrics and various optimization criteria. For we provide tight bounds, and for small we can obtain nontrivial
lower and upper bounds. Finally, we provide an algorithm to compute the
topological stability ratio in polynomial time for constant
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