A Local Algorithm for the Sparse Spanning Graph Problem

Constructing a sparse \emph{spanning subgraph} is a fundamental primitive in graph theory. In this paper, we study this problem in the Centralized Local model, where the goal is to decide whether an edge is part of the spanning subgraph by examining only a small part of the input; yet, answers must be globally consistent and independent of prior queries. Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees, cannot be constructed efficiently in this model. Therefore, we settle for a spanning subgraph containing at most $(1+\varepsilon)n$ edges (where $n$ is the number of vertices and $\varepsilon$ is a given approximation/sparsity parameter). We achieve query complexity of $\tilde{O}(poly(\Delta/\varepsilon)n^{2/3})$,\footnote{$\tilde{O}$-notation hides polylogarithmic factors in $n$.} where $\Delta$ is the maximum degree of the input graph. Our algorithm is the first to do so on arbitrary graphs. Moreover, we achieve the additional property that our algorithm outputs a \emph{spanner,} i.e., distances are approximately preserved. With high probability, for each deleted edge there is a path of $O(poly(\Delta/\varepsilon)\log^2 n)$ hops in the output that connects its endpoints

First NACO observations of the Brown Dwarf LHS 2397aB

Observations of the standard late type M8 star LHS 2397aA were obtained at the ESO-VLT 8m telescope ``Yepun'' using the NAOS/CONICA Adaptive Optics facility. The observations were taken during the NACO commissioning, and the infrared standard star LHS 2397aA was observed in the H, and Ks broad band filters. In both bands the brown dwarf companion LHS2397aB was detected. Using a program recently developed (Bouy et al., 2003) for the detection of stellar binaries we calculated the principal astrometric parameters (angular binary separation and position angle P.A.) and the photometry of LHS 2397aA and LHS 2397aB. Our study largely confirms previous results obtained with the AO-Hokupa'a facility at Gemini-North (Freed et al., 2003); however a few discrepancies are observed.Comment: 5 page

Mid-Infrared Instrumentation for the European Extremely Large Telescope

MIDIR is the proposed thermal/mid-IR imager and spectrograph for the European Extremely Large Telescope (E-ELT). It will cover the wavelength range of 3 to at least 20 microns. Designed for diffraction-limited performance over the entire wavelength range, MIDIR will require an adaptive optics system; a cryogenically cooled system could offer optimal performance in the IR, and this is a critical aspect of the instrument design. We present here an overview of the project, including a discussion of MIDIR's science goals and a comparison with other infrared (IR) facilities planned in the next decade; top level requirements derived from these goals are outlined. We describe the optical and mechanical design work carried out in the context of a conceptual design study, and discuss some important issues to emerge from this work, related to the design, operation and calibration of the instrument. The impact of telescope optical design choices on the requirements for the MIDIR instrument is demonstrated.Comment: for publication in SPIE Proceedings vol. 6692, Cryogenic Optical Systems and Instrumentation XII, eds. J.B. Heaney and L.G. Burriesci, San Diego, Aug 200

X-rays from Cepheus A East and West

We report the discovery of X-rays from both components of Cepheus A, East and West, with the XMM-Newton Observatory. HH 168 joins the ranks of other energetic Herbig Haro objects that are sources of T~10^6 K X-ray emission. The HH 168 effective temperature is T = 5.8 (+3.5,-2.3) x 10^6 K and its unabsorbed luminosity is 1.1 x 10^29 erg s^-1, making it hotter and less luminous than other representatives of its class. We also detect prominent X-ray emission from the complex of compact radio sources believed to be the power sources for Cep A. We call this source HWX and it is distinguished by its hard X-ray spectrum, T = 1.2 (+1.2,-0.5) x 10^8 K, and complex spatial distribution. It may arise from one or more protostars associated with the radio complex, the outflows, or a combination of the two. We detect 102 X-rays sources; many presumed to be pre-main sequence stars based upon the reddening of their optical/IR counterparts.Comment: 15 pages, 8 figures, data table not included because of size limit

Low diameter graph decompositions by approximate distance computation

In many models for large-scale computation, decomposition of the problem is key to efficient algorithms. For distance-related graph problems, it is often crucial that such a decomposition results in clusters of small diameter, while the probability that an edge is cut by the decomposition scales linearly with the length of the edge. There is a large body of literature on low diameter graph decomposition with small edge cutting probabilities, with all existing techniques heavily building on single source shortest paths (SSSP) computations. Unfortunately, in many theoretical models for large-scale computations, the SSSP task constitutes a complexity bottleneck. Therefore, it is desirable to replace exact SSSP computations with approximate ones. However this imposes a fundamental challenge since the existing constructions of low diameter graph decomposition with small edge cutting probabilities inherently rely on the subtractive form of the triangle inequality, which fails to hold under distance approximation. The current paper overcomes this obstacle by developing a technique termed blurry ball growing. By combining this technique with a clever algorithmic idea of Miller et al. (SPAA 2013), we obtain a construction of low diameter decompositions with small edge cutting probabilities which replaces exact SSSP computations by (a small number of) approximate ones. The utility of our approach is showcased by deriving efficient algorithms that work in the CONGEST, PRAM, and semi-streaming models of computation. As an application, we obtain metric tree embedding algorithms in the vein of Bartal (FOCS 1996) whose computational complexities in these models are optimal up to polylogarithmic factors. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is “used” only logaritmically many times, which is of interest for capacitated problems and simulating CONGEST algorithms on the tree into which the graph is embedded

Distributed algorithms for low stretch spanning trees

Given an undirected graph with integer edge lengths, we study the problem of approximating the distances in the graph by a spanning tree based on the notion of stretch. Our main contribution is a distributed algorithm in the CONGEST model of computation that constructs a random spanning tree with the guarantee that the expected stretch of every edge is O(log3 n), where n is the number of nodes in the graph. If the graph is unweighted, then this algorithm can be implemented to run in O(D) rounds, where D is the hop-diameter of the graph, thus being asymptotically optimal. In the weighted case, the run-time of our algorithm matches the currently best known bound for exact distance computations, i.e., Õ(min{√nD, √nD1/4 + n3/5 + D}). We stress that this is the first distributed construction of spanning trees leading to poly-logarithmic expected stretch with non-trivial running time

Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models

We present a method for solving the shortest transshipment problem-also known as uncapacitated minimum cost flow-up to a multiplicative error of 1 + ϵ in undirected graphs with non-negative integer edge weights using a tailored gradient descent algorithm. Our gradient descent algorithm takes ϵ-3 polylog n iterations, and in each iteration it needs to solve an instance of the transshipment problem up to a multiplicative error of polylog n, where n is the number of nodes. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. Using a careful white-box analysis, we can further extend the method to finding approximate solutions for the single-source shortest paths (SSSP) problem. As a consequence, we improve prior work by obtaining the following results: 1. Broadcast CONGEST model: (1+")-approximate SSSP using Õ((√ n+D) · ϵ-O(1)) rounds, 1 where D is the (hop) diameter of the network. 2. Broadcast congested clique model: (1+ϵ)-approximate shortest transshipment and SSSP using Õ (ϵ-O(1)) rounds. 3. Multipass streaming model: (1+ϵ)-approximate shortest transshipment and SSSP using Õ (n) space and Õ(ϵ-O(1)) passes. The previously fastest SSSP algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative integer edge weights that are polynomially bounded in n; for general nonnegative weights, running times scale with the logarithm of the maximum ratio between non-zero weights. In case of asymmetric costs for traversing an edge in opposite directions, running times scale with the maximum ratio between the costs of both directions over all edges

VLT Diffraction Limited Imaging and Spectroscopy in the NIR: Weighing the black hole in Centaurus A with NACO

We present high spatial resolution near-infrared spectra and images of the nucleus of Centaurus A (NGC 5128) obtained with NAOS-CONICA at the VLT. The adaptive optics corrected data have a spatial resolution of 0.06" (FWHM) in K- and 0.11" in H-band, four times higher than previous studies. The observed gas motions suggest a kinematically hot disk which is orbiting a central object and is oriented nearly perpendicular to the nuclear jet. We model the central rotation and velocity dispersion curves of the [FeII] gas orbiting in the combined potential of the stellar mass and the (dominant) black hole. Our physically most plausible model, a dynamically hot and geometrically thin gas disk, yields a black hole mass of M_bh = (6.1 +0.6/-0.8) 10^7 M_sun. As the physical state of the gas is not well understood, we also consider two limiting cases: first a cold disk model, which completely neglects the velocity dispersion; it yields an M_bh estimate that is almost two times lower. The other extreme case is to model a spherical gas distribution in hydrostatic equilibrium through Jeans equation. Compared to the hot disk model the best-fit black hole mass increases by a factor of 1.5. This wide mass range spanned by the limiting cases shows how important the gas physics is even for high resolution data. Our overall best-fitting black hole mass is a factor of 2-4 lower than previous measurements. With our revised M_bh estimate, Cen A's offset from the M_bh-sigma relation is significantly reduced; it falls above this relation by a factor of ~2, which is close to the intrinsic scatter of this relation. (Abridged)Comment: 12 pages, 14 figures, including minor changes following the referee report; accepted for publication in The Astrophysical Journa

An apodizing phase plate coronagraph for VLT/NACO

We describe a coronagraphic optic for use with CONICA at the VLT that provides suppression of diffraction from 1.8 to 7 lambda/D at 4.05 microns, an optimal wavelength for direct imaging of cool extrasolar planets. The optic is designed to provide 10 magnitudes of contrast at 0.2 arcseconds, over a D-shaped region in the image plane, without the need for any focal plane occulting mask.Comment: 9 pages, 5 figures, to appear in Proc. SPIE Vol. 773

A direct and differential imaging search for sub-stellar companions to epsilon Indi A

We have carried out a direct and differential imaging search for sub-stellar companions to eps Indi A using the adaptive optics system NACO at the ESO VLT. The observations were carried out in September 2004 with NACO/SDI as well as with NACO's S27 camera in the H and Ks filters. The SDI data cover an area of \~2.8" around eps Indi A. No detection was achieved in the inner neighbourhood down to 53 Mj (5 sigma confidence level) at a separation > 0.4" (1.45 AU) and down to 21 Mj for separations > 1.3" (4.7 AU). To cover a wider field of view, observations with the S27 camera and a coronagraphic mask were obtained. We detected a faint source at a separation of (7.3 +/- 0.1)" and a position angle of (302.9 +/- 0.8) degree. The photometry for the candidate companion yields m(H)=(16.45 +/- 0.04)mag and m(Ks) = (15.41 +/- 0.06)mag, respectively. Those magnitudes and the resulting color (H-Ks) = (1.04 +/- 0.07)mag fit best to a spectral type of L5 - L9.5 if it is bound. Observations done with HST/NICMOS by M. Endl have shown the source to be a background object.Comment: 6 pages, 5 figures, accepted by A&