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

作者：Bose Prosenjit; Devroye Luc; Loffler Maarten; Snoeyink Jack^*; Verma Vishal

来源：Computational Geometry-Theory and Applications, 2011, 44(2): 121-127.

DOI：10.1016/j.comgeo.2010.09.009

• Dilation
• Spanning ratio
• Shortest path
• Point configuration

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' is an element of P, which is the ratio of the length of the shortest path from p to p' in T over the Euclidean distance parallel to pp'parallel to. can be at most pi/2 approximate to 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.

• 出版日期2011-2