A tight lower bound instance for k-means++ in constant dimension

The k-means++ seeding algorithm is one of the most popular algorithms that is used for finding the initial $k$ centers when using the k-means heuristic. The algorithm is a simple sampling procedure and can be described as follows: Pick the first center randomly from the given points. For $i > 1$, pick a point to be the $i^{th}$ center with probability proportional to the square of the Euclidean distance of this point to the closest previously $(i-1)$ chosen centers. The k-means++ seeding algorithm is not only simple and fast but also gives an $O(\log{k})$ approximation in expectation as shown by Arthur and Vassilvitskii. There are datasets on which this seeding algorithm gives an approximation factor of $\Omega(\log{k})$ in expectation. However, it is not clear from these results if the algorithm achieves good approximation factor with reasonably high probability (say $1/poly(k)$). Brunsch and R\"{o}glin gave a dataset where the k-means++ seeding algorithm achieves an $O(\log{k})$ approximation ratio with probability that is exponentially small in $k$. However, this and all other known lower-bound examples are high dimensional. So, an open problem was to understand the behavior of the algorithm on low dimensional datasets. In this work, we give a simple two dimensional dataset on which the seeding algorithm achieves an $O(\log{k})$ approximation ratio with probability exponentially small in $k$. This solves open problems posed by Mahajan et al. and by Brunsch and R\"{o}glin.Comment: To appear in TAMC 2014. arXiv admin note: text overlap with arXiv:1306.420

Enhanced emission and light control with tapered plasmonic nanoantennas

We introduce a design of Yagi-Uda plasmonic nanoantennas for enhancing the antenna gain and achieving control over the angular emission of light. We demonstrate that tapering of antenna elements allows to decrease spacing between the antenna elements tenfold also enhancing its emission directivity. We find the optimal tapering angle that provides the maximum directivity enhancement and the minimum end-fire beamwidth

Communications of the lunar planetary laboratory, volume 3, no. 40- the system of lunar craters, quadrant ii

Observable properties of craters in second lunar quadran

Measurement of the Spatial Cross-Correlation Function of Damped Lyman Alpha Systems and Lyman Break Galaxies

We present the first spectroscopic measurement of the spatial cross-correlation function between damped Lyman alpha systems (DLAs) and Lyman break galaxies (LBGs). We obtained deep u'BVRI images of nine QSO fields with 11 known z ~ 3 DLAs and spectroscopically confirmed 211 R < 25.5 photometrically selected z > 2 LBGs. We find strong evidence for an overdensity of LBGs near DLAs versus random, the results of which are similar to that of LBGs near other LBGs. A maximum likelihood cross-correlation analysis found the best fit correlation length value of r_0 = 2.9^(+1.4)_(-1.5) h^(-1)Mpc using a fixed value of gamma = 1.6. The implications of the DLA-LBG clustering amplitude on the average dark matter halo mass of DLAs are discussed.Comment: 12 pages, 2 figures, accepted for publication in Astrophysical Journal Letter