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17
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 ..."
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Cited by 17 (10 self)
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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 ..."
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Cited by 15 (4 self)
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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. ..."
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Cited by 10 (8 self)
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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 ..."
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Cited by 8 (2 self)
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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 ..."
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Cited by 5 (3 self)
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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].
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 ..."
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Cited by 2 (1 self)
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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.
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 ..."
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Cited by 2 (1 self)
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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.
Improved lower bound on the geometric dilation of point sets
 Technische Universiteit Eindhoven
, 2005
"... 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 length of a shortest path connecting p and q in G divided by their Euclidean distance pq. The maximum detour over all pairs of points is called the geometric dilation ..."
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Cited by 1 (1 self)
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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 length of a shortest path connecting p and q in G divided by 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, a disk packing result and arguments from convex geometry, to prove this conjecture. The lower bound is improved to (1 + 10 −11)π/2. 1
Computing the detour and spanning ratio of paths, trees and cycles in 2D and 3D
"... 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 generalizi ..."
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Cited by 1 (1 self)
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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 obtain ���������������� �time 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.