Let S be a finite set of points in the Euclidean plane. Let D be a Delaunay triangulation of S. The stretch factor (also known as dilation or spanning ratio) of D is the maximum ratio, among all points p and q in S, of the shortest path distance from p to q in D over the Euclidean distance parallel to pq parallel to. Proving a tight bound on the stretch factor of the Delaunay triangulation has been a long-standing open problem in computational geometry. In this paper we prove that the stretch factor of the Delaunay triangulation is less than rho = 1.998, significantly improving the current best upper bound of 2.42 by Keil and Gutwin ["The Delaunay triangulation closely approximates the complete Euclidean graph," in Proceedings of the 1st Workshop on Algorithms and Data Structures (WADS), 1989, pp. 47-56]. Our bound of 1.998 also improves the upper bound of the best stretch factor that can be achieved by a plane spanner of a Euclidean graph (the current best upper bound is 2). Our result has a direct impact on the problem of constructing spanners of Euclidean graphs, which has applications in the area of wireless computing.
Title
The stretch factor of the Delaunay triangulation is less than 1.998