Optimum steiner ratio for gradient-constrained networks connecting three points in 3-space, part II: The gradient-constraint m satisfies 1 ≤ m ≤ √3

Abstract

The authors have proved in a previous article that the Steiner ratio (the minimum ratio of the length of a Steiner minimal tree to the length of a minimal spanning tree) for a gradient-constrained network on three points is 1/2+(√4+3m2-1)/(4√1+m2) if the maximum gradient m of an edge in the network satisfies m ≤ 1. In this article, we continue this study and use the same strategy to show that this result still holds when the maximum value of m satisfies 1 ≤ m ≤ √3. However the ratio becomes (3m+√9m2+12)/(4√3m2+3) if m ≥ √3. © 2010 Wiley Periodicals, Inc.