Bounds on the maximum multiplicity of some common geometric graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, non-weighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits {\Omega}(8.65^n) different triangulations. This improves the bound {\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by Aichholzer et al. (ii) We present a new lower bound of {\Omega}(12.00^n) for the number of non-crossing spanning trees of the double chain composed of two convex chains. The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30^n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62^n) non-crossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest non-crossing tours can be exponential in n. Likewise, we show that both the number of longest non-crossing tours and the number of longest non-crossing perfect matchings can be exponential in n. Moreover, we show that there are sets of n points in convex position with an exponential number of longest non-crossing spanning trees. For points in convex position we obtain tight bounds for the number of longest and shortest tours. We give a combinatorial characterization of the longest tours, which leads to an O(nlog n) time algorithm for computing them

An Overview of Rendering from Volume Data --- including Surface and Volume Rendering

Volume rendering is a title often ambiguously used in science. One meaning often quoted is: `to render any three volume dimensional data set'; however, within this categorisation `surface rendering'' is contained. Surface rendering is a technique for visualising a geometric representation of a surface from a three dimensional volume data set. A more correct definition of Volume Rendering would only incorporate the direct visualisation of volumes, without the use of intermediate surface geometry representations. Hence we state: `Volume Rendering is the Direct Visualisation of any three dimensional Volume data set; without the use of an intermediate geometric representation for isosurfaces'; `Surface Rendering is the Visualisation of a surface, from a geometric approximation of an isosurface, within a Volume data set'; where an isosurface is a surface formed from a cross connection of data points, within a volume, of equal value or density. This paper is an overview of both Surface Rendering and Volume Rendering techniques. Surface Rendering mainly consists of contouring lines over data points and triangulations between contours. Volume rendering methods consist of ray casting techniques that allow the ray to be cast from the viewing plane into the object and the transparency, opacity and colour calculated for each cell; the rays are often cast until an opaque object is `hit' or the ray exits the volume

Meeting in a Polygon by Anonymous Oblivious Robots

The Meeting problem for $k\geq 2$ searchers in a polygon $P$ (possibly with holes) consists in making the searchers move within $P$, according to a distributed algorithm, in such a way that at least two of them eventually come to see each other, regardless of their initial positions. The polygon is initially unknown to the searchers, and its edges obstruct both movement and vision. Depending on the shape of $P$, we minimize the number of searchers $k$ for which the Meeting problem is solvable. Specifically, if $P$ has a rotational symmetry of order $\sigma$ (where $\sigma=1$ corresponds to no rotational symmetry), we prove that $k=\sigma+1$ searchers are sufficient, and the bound is tight. Furthermore, we give an improved algorithm that optimally solves the Meeting problem with $k=2$ searchers in all polygons whose barycenter is not in a hole (which includes the polygons with no holes). Our algorithms can be implemented in a variety of standard models of mobile robots operating in Look-Compute-Move cycles. For instance, if the searchers have memory but are anonymous, asynchronous, and have no agreement on a coordinate system or a notion of clockwise direction, then our algorithms work even if the initial memory contents of the searchers are arbitrary and possibly misleading. Moreover, oblivious searchers can execute our algorithms as well, encoding information by carefully positioning themselves within the polygon. This code is computable with basic arithmetic operations, and each searcher can geometrically construct its own destination point at each cycle using only a compass. We stress that such memoryless searchers may be located anywhere in the polygon when the execution begins, and hence the information they initially encode is arbitrary. Our algorithms use a self-stabilizing map construction subroutine which is of independent interest.Comment: 37 pages, 9 figure

Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces

The Subgraph Isomorphism problem asks, given a host graph G on n vertices and a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P. The restriction of this problem to planar graphs has often been considered. After a sequence of improvements, the current best algorithm for planar graphs is a linear time algorithm by Dorn (STACS '10), with complexity $2^{O(k)} O(n)$. We generalize this result, by giving an algorithm of the same complexity for graphs that can be embedded in surfaces of bounded genus. At the same time, we simplify the algorithm and analysis. The key to these improvements is the introduction of surface split decompositions for bounded genus graphs, which generalize sphere cut decompositions for planar graphs. We extend the algorithm for the problem of counting and generating all subgraphs isomorphic to P, even for the case where P is disconnected. This answers an open question by Eppstein (SODA '95 / JGAA '99)

Textile Society of America Newsletter 28:1 — Spring 2016

Letter from the Editor Volunteer Opportunity: TSA Is Looking for a New Proceedings Editor Letter from the President Textiles Close Up Report: Art of the Zo: Textiles from Myanmar, India, and Bangladesh, Chin Weaving at the Philadelphia Museum of Art R. L. Shep Ethnic Textile Book Award 2015 Nominees Ossabaw Island, Indigo, and Sea Island Cotton: Two Ways to See a Georgia Barrier Island Peer-Review Process Yields Range of Exciting Exhibitions for Biennial Symposium Book Reviews:Symbols of Power: Luxury Items from Islamic Lands, 7th–21st CenturyTextiles of the Banjara: Cloth and Culture of a Wandering TribeThe Handbook of Textile CultureTraditional Weavers of Guatemala: Their Stories, Their LivesDesigning Identity: The Power of Textiles in Late Antiquity Conference Review: 21st Annual Weaving History Conference, 2015 Featured Exhibitions:Heirlooms, Catastrophe, and Survival: The Lace and Sampler Collection of the Palazzo DavanzatiThe Fabric of India, Victoria and Albert Museum, LondonFashion Meets Technology in #techstyle, Museum of Fine Arts, Boston Shibori and Ikat in Mesoamerica International Report:The Centre for Textile Conservation at the University of Glasgow & a New Era for Textile Dye Research in ScotlandThe Philippine Textile Research Institute, Taguig City, Philippines Member News:Member PublicationsMember Workshops and LecturesMember Awards & HonorsMember Exhibitions Conferences & Opportunities Symposium Program [Crosscurrents: Land, Labor, and the Port, 15th Biennial Symposium, Savannah, Georgia, October 19–23, 2016

Optimal area triangulation

Given a set of points in the Euclidean plane, we are interested in its triangulations, i.e., the maximal sets of non-overlapping triangles with vertices in the given points whose union is the convex hull of the point set. With respect to the area of the triangles in a triangulation, several optimality criteria can be considered. We study two of them. The MaxMin area triangulation is the triangulation of the point set that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the triangulation that minimizes the area of the largest area triangle in the triangulation. In the case when the point set is in a convex position, we present algorithms that construct MaxMin and MinMax area triangulations of a convex polygon in $O(n^2log{n})$ time and $O(n^2)$ space. These algorithms are based on dynamic programming. They use a number of geometric properties that are established within this work, and a variety of data structures specific to the problems. Further, we study polynomial time computable approximations to the optimal area triangulations of general point sets. We present geometric properties, based on angular constraints and perfect matchings, and use them to evaluate the approximation factor and to achieve triangulations with good practical quality compared to the optimal ones. These results open new direction in the research on optimal triangulations and set the stage for further investigations on optimization of area