A stochastic template placement algorithm for gravitational wave data analysis

This paper presents an algorithm for constructing matched-filter template banks in an arbitrary parameter space. The method places templates at random, then removes those which are "too close" together. The properties and optimality of stochastic template banks generated in this manner are investigated for some simple models. The effectiveness of these template banks for gravitational wave searches for binary inspiral waveforms is also examined. The properties of a stochastic template bank are then compared to the deterministically placed template banks that are currently used in gravitational wave data analysis.Comment: 14 pages, 11 figure

How dense can one pack spheres of arbitrary size distribution?

We present the first systematic algorithm to estimate the maximum packing density of spheres when the grain sizes are drawn from an arbitrary size distribution. With an Apollonian filling rule, we implement our technique for disks in 2d and spheres in 3d. As expected, the densest packing is achieved with power-law size distributions. We also test the method on homogeneous and on empirical real distributions, and we propose a scheme to obtain experimentally accessible distributions of grain sizes with low porosity. Our method should be helpful in the development of ultra-strong ceramics and high performance concrete.Comment: 5 pages, 5 figure

Precise Algorithm to Generate Random Sequential Addition of Hard Hyperspheres at Saturation

Random sequential addition (RSA) time-dependent packing process, in which congruent hard hyperspheres are randomly and sequentially placed into a system without interparticle overlap, is a useful packing model to study disorder in high dimensions. Of particular interest is the infinite-time {\it saturation} limit in which the available space for another sphere tends to zero. However, the associated saturation density has been determined in all previous investigations by extrapolating the density results for near-saturation configurations to the saturation limit, which necessarily introduces numerical uncertainties. We have refined an algorithm devised by us [S. Torquato, O. Uche, and F.~H. Stillinger, Phys. Rev. E {\bf 74}, 061308 (2006)] to generate RSA packings of identical hyperspheres. The improved algorithm produce such packings that are guaranteed to contain no available space using finite computational time with heretofore unattained precision and across the widest range of dimensions ($2 \le d \le 8$). We have also calculated the packing and covering densities, pair correlation function $g_2(r)$ and structure factor $S(k)$ of the saturated RSA configurations. As the space dimension increases, we find that pair correlations markedly diminish, consistent with a recently proposed "decorrelation" principle, and the degree of "hyperuniformity" (suppression of infinite-wavelength density fluctuations) increases. We have also calculated the void exclusion probability in order to compute the so-called quantizer error of the RSA packings, which is related to the second moment of inertia of the average Voronoi cell. Our algorithm is easily generalizable to generate saturated RSA packings of nonspherical particles

A note on the lower bound for online strip packing

This note presents a lower bound of $3/2+\sqrt{33}/6 \approx 2.457$ on the competitive ratio for online strip packing. The instance construction we use to obtain the lower bound was first coined by Brown, Baker and Katseff (1980). Recently this instance construction is used to improve the lower bound in computer aided proofs. We derive the best possible lower bound that can be obtained with this instance construction

Densest local packing diversity. II. Application to three dimensions

The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N = 1054. In the predecessor to this paper [A.B. Hopkins, F.H. Stillinger and S. Torquato, Phys. Rev. E 81 041305 (2010)], we described our method for finding the putative densest packings of N spheres in d-dimensional Euclidean space Rd and presented those packings in R2 for values of N up to N = 348. We analyze the properties and characteristics of the densest local packings in R3 and employ knowledge of the Rmin(N), using methods applicable in any d, to construct both a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. In R3, we find wide variability in the densest local packings, including a multitude of packing symmetries such as perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest local packings of N spheres near a central sphere to minimal-energy configurations of N+1 points interacting with short-range repulsive and long-range attractive pair potentials, e.g., 12-6 Lennard-Jones, and find that they are in general completely different, a result that has possible implications for nucleation theory. We also compare the densest local packings to finite subsets of stacking variants of the densest infinite packings in R3 (the Barlow packings) and find that the densest local packings are almost always most similar, as measured by a similarity metric, to the subsets of Barlow packings with the smallest number of coordination shells measured about a single central sphere, e.g., a subset of the FCC Barlow packing. We additionally observe that the densest local packings are dominated by the spheres arranged with centers at precisely distance Rmin(N) from the fixed sphere's center.Comment: 45 pages, 18 figures, 2 table

Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

We study the Parallel Task Scheduling problem $Pm|size_j|C_{\max}$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m \geq 5$, while it is solvable in pseudo-polynomial time for each $m \leq 3$. We give a positive answer to the long-standing open question whether this problem is strongly $NP$-complete for $m=4$. As a second result, we improve the lower bound of $\frac{12}{11}$ for approximating pseudo-polynomial Strip Packing to $\frac{5}{4}$. Since the best known approximation algorithm for this problem has a ratio of $\frac{4}{3} + \varepsilon$, this result narrows the gap between approximation ratio and inapproximability result by a significant step. Both results are proven by a reduction from the strongly $NP$-complete problem 3-Partition