30580193
A PTAS for minimum vertex dilation triangulation of a simple polygon with a constant number of sources of dilation
We design a fully polynomial-time approximation scheme for the problem of finding a triangulation of a simple polygon with a constant number of sources of dilation that achieves the minimum vertex dilation.
1585010
Exact and Approximation Algorithms for Computing the Dilation Spectrum of Paths, Trees, and Cycles
Let G be a graph embedded in Euclidean space. For any two vertices of G their dilation denotes the length of a shortest connecting path in G, divided by their Euclidean distance. In this paper we study the spectrum of the dilation, over all pairs of vertices of G. For paths, trees, and cycles in 2D we present O(n3/2+e) randomized algorithms that compute, for a given value κ ≥ 1, the exact number of vertex pairs of dilation > κ. Then we present deterministic algorithms that approximate the number of vertex pairs of dilation > κ up to an 1+η factor. They run in time O(n log2n) for chains and cycles, and in time O(n log3n) for trees, in any constant dimension.
696
Online searching with an autonomous robot
We discuss online strategies for visibility-based searching for an object hidden behind a corner, using Kurt3D, a real autonomous mobile robot. This task is closely related to a number of well-studied problems. Our robot uses a three-dimensional laser scanner in a stop, scan, plan, go fashion for building a virtual three-dimensional environment. Besides planning trajectories and avoiding obstacles, Kurt3D is capable of identifying objects like a chair. We derive a practically useful and asymptotically optimal strategy that guarantees a competitive ratio of 2, which differs remarkably from the well-studied scenario without the need of stopping for surveying the environment. Our strategy is used by Kurt3D, documented in a separate video.
6552276
On the geometric dilation of closed curves, graphs, and point sets
Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the ratio between the length of a shortest path connecting p and q in G and their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). 
 
Ebbers-Baumann, Grune and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation δ ≥ π/2 ≈ 1.57. They conjectured that the lower bound is not tight. 
 
We use new ideas like the halving pair transformation, a disk packing result and arguments from convex geometry, to prove this conjecture. The lower bound is improved to (1 + 10-11) π/2. The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance (h = H), examine the relation of h to other geometric quantities and prove some new dilation bounds.
8529966
Graphische Datenverarbeitung
3012905
The density of iterated crossing points and a gap result for triangulations of finite point sets
Consider a plane graph <i>G</i>, drawn with straight lines. For every pair <i>a,b</i> of vertices of <i>G</i>, we compare the shortest-path distance between <i>a</i> and <i>b</i> in <i>G</i> (with Euclidean edge lengths) to their actual distance in the plane. The worst-case ratio of these two values, for all pairs of points, is called the <i>dilation</i> of <i>G</i>. All finite plane graphs of dilation 1 have been classified. They are closely related to the following iterative procedure. For a given point set <i>P</i> ⊆ <i>R</i><sup>2</sup>, we connect every pair of points in <i>P</i> by a line segment and then add to <i>P</i> all those points where two such line segments cross. Repeating this process infinitely often, yields a limit point set <i>P</i><sup>∞</sup>⊇<i>P</i>. This limit set <i>P</i><sup>∞</sup> is finite if and only if <i>P</i> is contained in the vertex set of a triangulation of dilation 1.The main result of this paper is the following gap theorem: For any finite point set <i>P</i> in the plane for which <i>P</i><sup>∞</sup> is infinite, there exists a threshold λ > 1 such that <i>P</i> is not contained in the vertex set of any finite plane graph of dilation at most λ. As a first ingredient to our proof, we show that such an infinite <i>P</i><sup>∞</sup> must lie dense in a certain region of the plane. In the second, more difficult part, we then construct a concrete point set <i>P</i><sub>0</sub> such that any planar graph that contains this set amongst its vertices must have a dilation larger than 1.0000047.
18735413
Service and commodity based electronic intermediaries: a comparative analysis
The theoretical literature concerning supply chain intermediation is surveyed to develop a framework of five intermediary states. Data collected on 134 electronic intermediaries are employed to explore correlations between intermediation states, growth strategies and firm profitability, in a sample representing both service based and commodity based intermediaries. The study finds a positive relationship between more capital intensive states of intermediation and profitability. In addition, we find a higher use of acquisitions as growth mechanisms in service based intermediaries, and a greater use of joint ventures as growth mechanisms within commodity based portals.