Steiner trees in general nonuniform orientations

Abstract

In this paper we consider Steiner minimum trees (SMT) in the plane, where the connections can only be along a given set of fixed but arbitrary (not necessarily uniform) orientations. The orientations define a metric, called the general orientation metric, Aσ, where σ is the number of orientations. We prove that in Aσ metric, there exists an SMT whose Steiner points belong to an (n - 2)-level grid. This result generalizes a result by Lee and Shen [11], and a result by Du and Hwang [5]. In the former case, the same result was obtained for the special case when all orientations are uniform, while in the latter case the same result was proven for the special case when there are only three arbitrary orientations. We then modify the proof used in the main result for the special case when σ = 3, i.e., only three arbitrary orientations are considered, and obtain a better result, which states that there exists an SMT whose Steiner points belong to an [n-1/2]-level grid. The result has also been obtained by Lin and Xue [9] using a different approach.