Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity

We describe a linear-time algorithm that finds a planar drawing of every graph of a simple line or pseudoline arrangement within a grid of area O(n^{7/6}). No known input causes our algorithm to use area \Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would represent significant progress on the famous k-set problem from discrete geometry. Drawing line arrangement graphs is the main task in the Planarity puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing, Bordeaux, 201

The Price of Order

We present tight bounds on the spanning ratio of a large family of ordered $\theta$-graphs. A $\theta$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $\theta = 2 \pi/m$. An ordered $\theta$-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer $k \geq 1$, ordered $\theta$-graphs with $4k + 4$ cones have a tight spanning ratio of $1 + 2 \sin(\theta/2) / (\cos(\theta/2) - \sin(\theta/2))$. We also show that for any integer $k \geq 2$, ordered $\theta$-graphs with $4k + 2$ cones have a tight spanning ratio of $1 / (1 - 2 \sin(\theta/2))$. We provide lower bounds for ordered $\theta$-graphs with $4k + 3$ and $4k + 5$ cones. For ordered $\theta$-graphs with $4k + 2$ and $4k + 5$ cones these lower bounds are strictly greater than the worst case spanning ratios of their unordered counterparts. These are the first results showing that ordered $\theta$-graphs have worse spanning ratios than unordered $\theta$-graphs. Finally, we show that, unlike their unordered counterparts, the ordered $\theta$-graphs with 4, 5, and 6 cones are not spanners

Upper and Lower Bounds for Competitive Online Routing on Delaunay Triangulations

Consider a weighted graph G where vertices are points in the plane and edges are line segments. The weight of each edge is the Euclidean distance between its two endpoints. A routing algorithm on G has a competitive ratio of c if the length of the path produced by the algorithm from any vertex s to any vertex t is at most c times the length of the shortest path from s to t in G. If the length of the path is at most c times the Euclidean distance from s to t, we say that the routing algorithm on G has a routing ratio of c.We present an online routing algorithm on the Delaunay triangulation with competitive and routing ratios of 5.90. This improves upon the best known algorithm that has competitive and routing ratio 15.48. The algorithm is a generalization of the deterministic 1-local routing algorithm by Chew on the L1-Delaunay triangulation. When a message follows the routing path produced by our algorithm, its header need only contain the coordinates of s and t. This is an improvement over the currently known competitive routing algorithms on the Delaunay triangulation, for which the header of a message must additionally contain partial sums of distances along the routing path.We also show that the routing ratio of any deterministic k-local algorithm is at least 1.70 for the Delaunay triangulation and 2.70 for the L1-Delaunay triangulation. In the case of the L1-Delaunay triangulation, this implies that even though there exists a path between two points x and y whose length is at most 2.61|[xy]| (where |[xy]| denotes the length of the line segment [xy]), it is not always possible to route a message along a path of length less than 2.70|[xy]|. From these bounds on the routing ratio, we derive lower bounds on the competitive ratio of 1.23 for Delaunay triangulations and 1.12 for L1-Delaunay triangulations

Techniques for Restoration of Disturbed Coastal Wetlands of the Great Lakes

A long history of human-induced degradation of Great Lakes wetlands has made restoration a necessity, but the practice of wetland restoration is relatively new, especially in large lake systems. Therefore, we compiled tested methods and developed additional potential methods based on scientific understanding of Great Lakes wetland ecosystems to provide an overview of approaches fur restoration. We addressed this challenge by focusing on four general fields of science: hydrology, sedimentology, chemistry, and biology. Hydrologic remediation methods include restoring hydrologic connections between diked and hydrologically altered wetlands and the lakes, restoring water tables lowered by ditching, and restoring natural variation in lake levels of regulated lakes Superior and Ontario. Sedimentological remediation methods include management of sediment input from uplands, removal or proper management of dams on tributary rivers. and restoration of protective barrier beaches and sand spits. Chemical remediation methods include reducing or eliminating inputs of contaminants from point and non-point sources, natural sediment remediation by biodegradation and chemical degradation, and active sediment remediation by removal or by in situ treatment. Biological remediation methods include control of non-target organisms, enhancing populations of target organisms, and enhancing habitat for target organisms. Some of these methods were used in three major restoration projects (Metzger Marsh on Lake Erie and Cootes Paradise and Oshawa Second Marsh on Lake Ontario). which are described as case studies to show practical applications of wetland restoration in the Great Lakes. Successful restoration techniques that do not require continued manipulation must be founded in the basic tenets of ecology and should mimic natural processes. Success is demonstrated by the sustainability, productivity, nutrient-retention ability, invasibility, and biotic interactions within a restored wetland

The floodlight problem

Given three angles summing to 2, given n points in the plane and a tripartition k1 + k2 + k3 = n, we can tripartition the plane into three wedges of the given angles so that the i-th wedge contains ki of the points. This new result on dissecting point sets is used to prove that lights of specied angles not exceeding can be placed at n xed points in the plane to illuminate the entire plane if and only if the angles sum to at least 2. We give O(n log n) algorithms for both these problems

From Pariah To Partner-Russian-American Security Cooperation In The Artic Ocean

The Obama administration seeks to push the reset button on Washington-Moscow relations

Robust Geometric Spanners

Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in addition, geometric spanners. We define a property of spanners called robustness. Informally, when one removes a few vertices from a robust spanner, this harms only a small number of other vertices. We show that robust spanners must have a superlinear number of edges, even in one dimension. On the positive side, we give constructions, for any dimension, of robust spanners with a near-linear number of edges.Comment: 18 pages, 8 figure

Network Engineering through Skip List Data Structure for Federated Cloud Data Center

Now a days we can see that Cloud computing is fastest growing platform for making resources available to the users CC is based on internet computing that is enormous group of remote servers are network that allow centralized the data store and access. In terms of cloud computing this background network is known as Data Center Network. In this paper we have explain Skip List Topology because this topology in the account as the size of list is dynamic and it is very easy to insert and searching data in the list it is apparent that Skip List is better in term of efficiency B-tree