The Steiner Tree Problem in Orientation Metrics

Given a set of i (i = 1; 2; : : :; k) orientations (angles) in the plane, one can deene a distance function which induces a metric in the plane, called the orientation metric 3]. In the special case where all the angles are equal, we call the metric a uniform orientation metric 2]. Speciically, if there are orientations, forming angles ii ; 0 i ?1, with the x-axis, where 2 is an integer, we call the metric-metric. Note that the 2-metric is the well-known rectilinear metric and the 1 corresponds to the Euclidean metric. In this paper, we will concentrate on the 3-metric. In the 2-metric, Hanan 1] shows that there exists a solution of the Steiner tree problem such that all Steiner points are on the intersections of grid lines formed by passing lines at directions ii 2 ; i = 0; 1, through all demand points. But this is not true in the 3-metric. In this paper, we mainly prove the following theorem: be the set of n demand points, the set of Steiner points and the set of the ith generation intersection points, respectively. Then there exists a solution G of the Steiner problem S n such that for n = 3, all Steiner points are in O 1 and for n 4, they are in S k i=1 O i , where k d n?2 2 e.