Results 1  10
of
21
The Geometric Dilation of Finite Point Sets
, 2006
"... 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. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set, we would ..."
Abstract

Cited by 18 (10 self)
 Add to MetaCart
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. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph that contains the given points. In this paper we prove that a dilation of 1.678 is always sufficient, and that #/2 1.570 ... is sometimes necessary in order to accommodate a finite set of points.
Minimum dilation stars
 In Proc. ACM Symposium on Computational Geometry
, 2005
"... Abstract. The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in IR d. In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(nlog n)time algorit ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
Abstract. The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in IR d. In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(nlog n)time algorithm for evaluating the dilation of a given star; a randomized O(n log n) expectedtime algorithm for finding an optimal center in IR d; and for the case d = 2, a randomized O(nα(n) log 2 n) expectedtime algorithm for finding an optimal center among the input points. 1
Geometric dilation of closed planar curves: A new lower bound
, 2004
"... Given any simple closed curve C in the Euclidean plane, let w and D denote the minimal and the maximal caliper distances of C, correspondingly. We show that any such curve C has a geometric dilation of at least arcsin( D ) + ( w ) 1. ..."
Abstract

Cited by 10 (8 self)
 Add to MetaCart
Given any simple closed curve C in the Euclidean plane, let w and D denote the minimal and the maximal caliper distances of C, correspondingly. We show that any such curve C has a geometric dilation of at least arcsin( D ) + ( w ) 1.
Finding the best shortcut in a geometric network
 ACM Symp. Comput. Geom
, 2005
"... Given a Euclidean graph G in R d with n vertices and m edges we consider the problem of adding a shortcut such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a Euclidean graph runs in O(mn + n 2 log n) time, resulting ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
Given a Euclidean graph G in R d with n vertices and m edges we consider the problem of adding a shortcut such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a Euclidean graph runs in O(mn + n 2 log n) time, resulting in a trivial O(mn 3 + n 4 log n) time algorithm for computing the optimal shortcut. First, we show that a simple modification yields the optimal solution in O(n 4) time using O(n 2) space. To reduce the running times we consider several approximation algorithms. Our main result is a (2 + ε)approximation algorithm with running time O(nm + n 2 (log n +1/ε 3d)) using O(n 2)space.
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 geometri ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
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). EbbersBaumann, Grüne 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. Key words: computational geometry, convex geometry, convex curves, dilation, distortion, detour, lower bound, halving chord, halving pair, Zindler curves ∗Some results of this article were presented at the 21st European Workshop on Computational Geometry (EWCG ’05)[7], others at the 9th Workshop on Algorithms and Data Structures (WADS ’05)[6].
Light Orthogonal Networks with Constant Geometric Dilation
, 2008
"... An orthogonal spanner network for a given set of n points in the plane is a plane straight line graph with axisaligned edges that connects all input points. We show that for any set of n points in the plane, there is an orthogonal spanner network that (i) is short having a total edge length of at m ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
An orthogonal spanner network for a given set of n points in the plane is a plane straight line graph with axisaligned edges that connects all input points. We show that for any set of n points in the plane, there is an orthogonal spanner network that (i) is short having a total edge length of at most a constant times the length of a Euclidean minimum spanning tree for the point set; (ii) is small having O(n) vertices and edges; and (iii) has constant geometric dilation, which means that for any two points u and v in the network, the shortest path in the network between u and v is at most a constant times longer than the (Euclidean) distance between u and v. Such a network can be constructed in O(n log n) time.
On the geometric dilation of curves and point sets
, 2004
"... Let G be an embedded planar graph whose edges are curves. The detour between two points u and v (on edges or vertices) of G is the ratio between the shortest path in G between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebb ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Let G be an embedded planar graph whose edges are curves. The detour between two points u and v (on edges or vertices) of G is the ratio between the shortest path in G between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation δ(G). EbbersBaumann, Grüne and Klein have recently 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. We prove a stronger lower bound δ ≥ (1 + 10 −11)π/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks. 1
Computing the detour and spanning ratio of paths, trees and cycles in 2d and 3d. Discrete & Computational Geometry
"... The detour and spanning ratio of a graph embedded in measure how well approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we ob ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
The detour and spanning ratio of a graph embedded in measure how well approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we obtaintime algorithms for computing the detour or spanning ratio of planar trees and cycles. Finally, we develop subquadratic algorithms for computing the detour and spanning ratio for paths, cycles, and trees embedded! in, and show that computing the detour "! in is at least as hard as Hopcroft’s problem.
Exact and Approximation Algorithms for Computing the Dilation Spectrum of Paths, Trees, and Cycles
, 2005
"... 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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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(n 3/2+ɛ) 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 log 2 n) for chains and cycles, and in time O(n log 3 n) for trees, in any constant dimension.