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We describe an algorithm that allows one to find dense packing configurations of a number of congruent disks in arbitrary domains in two or more dimensions. We have applied it to a large class of two dimensional domains such as rectangles,…
Finding the densest sphere packing in $d$-dimensional Euclidean space $\mathbb{R}^d$ is an outstanding fundamental problem with relevance in many fields, including the ground states of molecular systems, colloidal crystal structures, coding…
We consider a multi-stage facility reallocation problems on the real line, where a facility is being moved between time stages based on the locations reported by $n$ agents. The aim of the reallocation algorithm is to minimise the social…
A sphere is a fundamental geometric object widely used in (computer aided) geometric design. It possesses rational parameterizations but no parametric polynomial parameterization exists. The present study provides an approach to the optimal…
We develop an algorithm to construct new self-similar space-filling packings of spheres. Each topologically different configuration is characterized by its own fractal dimension. We also find the first bi-cromatic packing known up to now.
We consider a variant of the classical Bin Packing Problem, called Fully Dynamic Bin Packing. In this variant, items of a size in $(0,1]$ must be packed in bins of unit size. In each time step, an item either arrives or departs from the…
In the \textsc{2-Dimensional Knapsack} problem (2DK) we are given a square knapsack and a collection of $n$ rectangular items with integer sizes and profits. Our goal is to find the most profitable subset of items that can be packed…
Fast quantum algorithms can solve important computational problems more efficiently than classical algorithms. However, little is known about whether quantum computing can speed up solving geometric problems. This article explores quantum…
Mixed packing and covering problems are problems that can be formulated as linear programs using only non-negative coefficients. Examples include multicommodity network flow, the Held-Karp lower bound on TSP, fractional relaxations of set…
Despite its long history, there are many fundamental issues concerning random packings of spheres that remain elusive, including a precise definition of random close packing (RCP). We argue that the current picture of RCP cannot be made…
Spatial queries like range queries, nearest neighbor, circular range queries etc. are the most widely used queries in the location-based applications. Building secure and efficient solutions for these queries in the cloud computing…
Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are…
The problem of adaptively setting the timeout interval for retransmitting a packet has been discussed. A layered view of the algorithms has been presented. It is shown that a timeout algorithm consists of essentially five layers or…
For the accurate representation and reconstruction of band-limited signals on the sphere, an optimal-dimensionality sampling scheme has been recently proposed which requires the optimal number of samples equal to the number of degrees of…
We develop a model to describe the properties of random assemblies of polydisperse hard spheres. We show that the key features to describe the system are (i) the dependence between the free volume of a sphere and the various coordination…
In this paper we propose an accurate, and computationally efficient method for incorporating adaptive spatial resolution into weakly-compressible Smoothed Particle Hydrodynamics (SPH) schemes. Particles are adaptively split and merged in an…
We use computational experiments to find the rectangles of minimum perimeter into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. In many of the packings…
We investigate equal spheres packings generated from several experiments and from a large number of different numerical simulations. The structural organization of these disordered packings is studied in terms of the network of common…
In this paper, we introduce optimal versions of a multi-port based teleportation scheme allowing to send a large amount of quantum information. We fully characterise probabilistic and deterministic case by presenting expressions for the…
This is the sixth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…