Lower bounds on the dilation of plane spanners

(I) We exhibit a set of 23 points in the plane that has dilation at least $1.4308$, improving the previously best lower bound of $1.4161$ for the worst-case dilation of plane spanners. (II) For every integer $n\geq13$, there exists an $n$-element point set $S$ such that the degree 3 dilation of $S$ denoted by $\delta_0(S,3) \text{ equals } 1+\sqrt{3}=2.7321\ldots$ in the domain of plane geometric spanners. In the same domain, we show that for every integer $n\geq6$, there exists a an $n$-element point set $S$ such that the degree 4 dilation of $S$ denoted by $\delta_0(S,4) \text{ equals } 1 + \sqrt{(5-\sqrt{5})/2}=2.1755\ldots$ The previous best lower bound of $1.4161$ holds for any degree. (III) For every integer $n\geq6$, there exists an $n$-element point set $S$ such that the stretch factor of the greedy triangulation of $S$ is at least $2.0268$.Comment: Revised definitions in the introduction; 23 pages, 15 figures; 2 table

EMBEDDING POINT SETS INTO PLANE GRAPHS OF SMALL DILATION

Let S be a set of points in the plane. What is the minimum possible dilation of all plane graphs that contain S? Even for a set S as simple as five points evenly placed on the circle, this question seems hard to answer; it is not even clear if there exists a lower bound> 1. In this paper we provide the first upper and lower bounds for the embedding problem. 1. Each finite point set can be embedded into the vertex set of a finite triangulation of dilation ≤ 1.1247. 2. Each embedding of a closed convex curve has dilation ≥ 1.00157. 3. Let P be the plane graph that results from intersecting n infinite families of equidistant, parallel lines in general position. Then the vertex set of P has dilation ≥ 2 / √ 3 ≈ 1.1547

A fast algorithm for approximating the detour of a polygonal chain

Let C be a simple(1) polygonal chain of n edges in the plane, and let p and q be two arbitrary points on C. The detour of C on (p, q) is defined to be the length of the subchain of C that connects p with q, divided by the Euclidean distance between p and q. Given an epsilon >0, we compute in time O((1)/(epsilon) n log n) a pair of points on which the chain makes a detour at least 1/(1 + epsilon) times the maximum detour. (C) 2003 Elsevier B.V. All rights reserved

Computing the Maximum Detour of a Plane Graph in Subquadratic Time

Let G be a plane graph where each edge is a line segment. We consider the problem of computing the maximum detour of G, defined as the maximum over all pairs of distinct points p and q of G of the ratio between the distance between p and q in G and the distance |pq|. The fastest known algorithm for this problem has Θ(n 2) running time where n is the number of vertices. We show how to obtain O(n 3/2 log 3 n) expected running time. We also show that if G has bounded treewidth, its maximum detour can be computed in O(n log 3 n) expected time

On the Geometric Dilation of Finite Point Sets

Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance

Computing the detour and spanning ratio of paths, trees, and cycles in 2D and 3D

SCOPUS: cp.jinfo:eu-repo/semantics/publishe