The Geometric Dilation of Finite Point Sets

Authors:
Annette Ebbers-Baumann;Ansgar Grune;Rolf Klein
Affiliations:
Universitat Bonn, Institut fur Informatik I, D-53117 Bonn, Germany;Universitat Bonn, Institut fur Informatik I, D-53117 Bonn, Germany;Universitat Bonn, Institut fur Informatik I, D-53117 Bonn, Germany
Venue:
Algorithmica
Year:
2006

Citing 0
Cited 11

The density of iterated crossing points and a gap result for triangulations of finite point sets

Proceedings of the twenty-second annual symposium on Computational geometry
On the geometric dilation of closed curves, graphs, and point sets

Computational Geometry: Theory and Applications - Special issue on the 21st European workshop on computational geometry (EWCG 2005)
Geometric dilation of closed planar curves: New lower bounds

Computational Geometry: Theory and Applications
Sparse geometric graphs with small dilation

Computational Geometry: Theory and Applications
Feed-links for network extensions

Proceedings of the 16th ACM SIGSPATIAL international conference on Advances in geographic information systems
Light orthogonal networks with constant geometric dilation

Journal of Discrete Algorithms
Connect the Dot: Computing Feed-Links with Minimum Dilation

WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Computing geometric minimum-dilation graphs is NP-hard

GD'06 Proceedings of the 14th international conference on Graph drawing
Light orthogonal networks with constant geometric dilation

STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Almost all Delaunay triangulations have stretch factor greater than π/2

Computational Geometry: Theory and Applications
Self-approaching graphs

GD'12 Proceedings of the 20th international conference on Graph Drawing

Quantified Score

Hi-index	0.00

Visualization

Abstract

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 π/2 = 1.570... is sometimes necessary in order to accommodate a finite set of points.