The Steiner Minimal Tree problem in the λ-geometry plane

A Steiner Minimal Tree (SMT) for a given set P of points is a shortest network interconnecting the points of P whose vertex set may include some additional points in order to get the minimum possible total length in a metric space. When no additional points are allowed the minimum interconnection network is the well-known minimum spanning tree (MST) of P. The Steiner ratio is the greatest lower bound of the ratio of the length of an SMT over that of an MST of P. In this paper we study the Steiner minimal tree problem in which all the edges of SMT have fixed orientations. We call it the SMT problem in the lambda-geometry plane, where lambda is the number of possible orientations. Here is the summary of our results. 1. We show that the Steiner ratio for \P\ greater than or equal to 3 is root 3/2 cos(pi/2 lambda), for lambda = 6m + 3 and integer m greater than or equal to 0, and is root 3/2, for lambda = 6k and integer k greater than or equal to 1, disproving a a conjecture of Du et al.[3] that the ratio is root 3/2 iff the unit disk in normed planes is an ellipse. 2. We derive the Steiner ratios for \P\ less than or equal to 4 for all possible lambda's and show that for \P\ greater than or equal to 3 there exists an SMT whose Steiner points lie in a multi-level Hanan-grid, generalizing a result that holds for rectilinear case, i.e., lambda = 2. These results show that the Steiner ratio is not a monotonically increasing function of lambda, as believed by many researchers. We conjecture that the Steiner ratios obtained above (\P\ less than or equal to 4) are actually true for all \P\ greater than or equal to 3.