Almost all Delaunay triangulations have stretch factor greater than π/2

Authors:
Prosenjit Bose;Luc Devroye;Maarten Löffler;Jack Snoeyink;Vishal Verma
Affiliations:
School of Computer Science, Carleton University, 5302 Herzberg Laboratories, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada;School of Computer Science, McGill University, 3480 University Street, Montreal, Canada H3A 2K6;Department of Information and Computing Sciences, Utrecht University, The Netherlands;UNC Computer Science, Chapel Hill, NC 27599-3175, USA;UNC Computer Science, Chapel Hill, NC 27599-3175, USA
Venue:
Computational Geometry: Theory and Applications
Year:
2011

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Abstract

Consider the Delaunay triangulation T of a set P of points in the plane as a Euclidean graph, in which the weight of every edge is its length. It has long been conjectured that the stretch factor in T of any pair p,p^'@?P, which is the ratio of the length of the shortest path from p to p^' in T over the Euclidean distance @?pp^'@?, can be at most @p/2~1.5708. In this paper, we show how to construct point sets in convex position with stretch factor 1.5810 and in general position with stretch factor 1.5846. Furthermore, we show that a sufficiently large set of points drawn independently from any distribution will in the limit approach the worst-case stretch factor for that distribution.