A strip-like tiling algorithm

AbstractWe extend our previous results on the connection between strip tiling problems and regular grammars by showing that an analogous algorithm is applicable to other tiling problems, not necessarily related to rectangular strips. We find generating functions for monomer and dimer tilings of T- and L-shaped figures, holed and slotted strips, diagonal strips and combinations of them, and show how analogous results can be obtained by using different pieces

The Tiling Algorithm for the 6dF Galaxy Survey

The Six Degree Field Galaxy Survey (6dFGS) is a spectroscopic survey of the southern sky, which aims to provide positions and velocities of galaxies in the nearby Universe. We present here the adaptive tiling algorithm developed to place 6dFGS fields on the sky, and allocate targets to those fields. Optimal solutions to survey field placement are generally extremely difficult to find, especially in this era of large-scale galaxy surveys, as the space of available solutions is vast (2N dimensional) and false optimal solutions abound. The 6dFGS algorithm utilises the Metropolis (simulated annealing) method to overcome this problem. By design the algorithm gives uniform completeness independent of local density, so as to result in a highly complete and uniform observed sample. The adaptive tiling achieves a sampling rate of approximately 95%, a variation in the sampling uniformity of less than 5%, and an efficiency in terms of used fibres per field of greater than 90%. We have tested whether the tiling algorithm systematically biases the large-scale structure in the survey by studying the two-point correlation function of mock 6dF volumes. Our analysis shows that the constraints on fibre proximity with 6dF lead to under-estimating galaxy clustering on small scales (< 1 Mpc) by up to ~20%, but that the tiling introduces no significant sampling bias at larger scales.Comment: 11 pages, 7 figures. Full resolution version of the paper available from http://www.mso.anu.edu.au/6dFGS/ . Abridged version of abstract belo

Random numbers from the tails of probability distributions using the transformation method

The speed of many one-line transformation methods for the production of, for example, Levy alpha-stable random numbers, which generalize Gaussian ones, and Mittag-Leffler random numbers, which generalize exponential ones, is very high and satisfactory for most purposes. However, for the class of decreasing probability densities fast rejection implementations like the Ziggurat by Marsaglia and Tsang promise a significant speed-up if it is possible to complement them with a method that samples the tails of the infinite support. This requires the fast generation of random numbers greater or smaller than a certain value. We present a method to achieve this, and also to generate random numbers within any arbitrary interval. We demonstrate the method showing the properties of the transform maps of the above mentioned distributions as examples of stable and geometric stable random numbers used for the stochastic solution of the space-time fractional diffusion equation.Comment: 17 pages, 7 figures, submitted to a peer-reviewed journa

The Homogeneous Broadcast Problem in Narrow and Wide Strips

Let $P$ be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let $s\in P$ be a given source node. Each node $p$ can transmit information to all other nodes within unit distance, provided $p$ is activated. The (homogeneous) broadcast problem is to activate a minimum number of nodes such that in the resulting directed communication graph, the source $s$ can reach any other node. We study the complexity of the regular and the hop-bounded version of the problem (in the latter, $s$ must be able to reach every node within a specified number of hops), with the restriction that all points lie inside a strip of width $w$. We almost completely characterize the complexity of both the regular and the hop-bounded versions as a function of the strip width $w$.Comment: 50 pages, WADS 2017 submissio