On compact packings of the plane with circles of three radii

A compact circle-packing $P$ of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so that, for every $i\in\{0,\ldots,n-1\}$, the circle $A_{i}$ is tangent to both circles $S$ and $A_{i+1\mod n}.$ We show that there exist at most $13617$ pairs $(r,s)$ with $0<s<r<1$ for which there exist a compact circle-packing of the plane consisting of circles with radii $s$, $r$ and $1$. We discuss computing the exact values of such $0<s<r<1$ as roots of polynomials and exhibit a selection of compact circle-packings consisting of circles of three radii. We also discuss the apparent infeasibility of computing \emph{all} these values on contemporary consumer hardware with the methods employed in this paper.Comment: Dataset referred to in the text can be obtained at http://dx.doi.org/10.17632/t66sfkn5tn.

An average-case analysis of bin packing with uniformly distributed item sizes

We analyze the one-dimensional bin-packing problem under the assumption that bins have unit capacity, and that items to be packed are drawn from a uniform distribution on [0,1]. Building on some recent work by Frederickson, we give an algorithm which uses n/2+0(n^½) bins on the average to pack n items. (Knodel has achieved a similar result.) The analysis involves the use of a certain 1-dimensional random walk. We then show that even an optimum packing under this distribution uses n/2+0(n^1/2) bins on the average, so our algorithm is asymptotically optimal, up to constant factors on the amount of wasted space. Finally, following Frederickson, we show that two well-known greedy bin-packing algorithms use no more bins than our algorithm; thus their behavior is also in asymptotically optimal in this sense

A Fast Distributed Stateless Algorithm for α\alpha-Fair Packing Problems

Over the past two decades, fair resource allocation problems have received considerable attention in a variety of application areas. However, little progress has been made in the design of distributed algorithms with convergence guarantees for general and commonly used $\alpha$-fair allocations. In this paper, we study weighted $\alpha$-fair packing problems, that is, the problems of maximizing the objective functions (i) $\sum_j w_j x_j^{1-\alpha}/(1-\alpha)$ when $\alpha > 0$, $\alpha \neq 1$ and (ii) $\sum_j w_j \ln x_j$ when $\alpha = 1$, over linear constraints $Ax \leq b$, $x\geq 0$, where $w_j$ are positive weights and $A$ and $b$ are non-negative. We consider the distributed computation model that was used for packing linear programs and network utility maximization problems. Under this model, we provide a distributed algorithm for general $\alpha$ that converges to an $\varepsilon-$approximate solution in time (number of distributed iterations) that has an inverse polynomial dependence on the approximation parameter $\varepsilon$ and poly-logarithmic dependence on the problem size. This is the first distributed algorithm for weighted $\alpha-$fair packing with poly-logarithmic convergence in the input size. The algorithm uses simple local update rules and is stateless (namely, it allows asynchronous updates, is self-stabilizing, and allows incremental and local adjustments). We also obtain a number of structural results that characterize $\alpha-$fair allocations as the value of $\alpha$ is varied. These results deepen our understanding of fairness guarantees in $\alpha-$fair packing allocations, and also provide insight into the behavior of $\alpha-$fair allocations in the asymptotic cases $\alpha\rightarrow 0$, $\alpha \rightarrow 1$, and $\alpha \rightarrow \infty$.Comment: Added structural results for asymptotic cases of \alpha-fairness (\alpha approaching 0, 1, or infinity), improved presentation, and revised throughou