The Steiner tree problem in orientation metrics

Given a set Theta of alpha(i) (i = 1, 2, ..., k) orientations (angles) in the plane, one can define 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]. Specifically, if there are a orientations, forming angles i pi/sigma, 0 less than or equal to i less than or equal to sigma - 1, with the x-axis, where sigma greater than or equal to 2 is an integer, we call the metric a lambda(sigma)-metric. Note that the lambda(2)-metric is the well-known rectilinear metric and the lambda(infinity) corresponds to the Euclidean metric. In this paper, we will concentrate on the lambda(3)-metric. In the lambda(2)-metric, Hanan has shown 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 i pi/2, i = 0, 1, through all demand points. But this is not true in the lambda(3)-metric. In this paper, we mainly prove the following theorem: Let P, Q, and O-i (i = 1, 2, ..., k) 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 all Steiner points are in boolean (ORi=1Oi)-O-k, where k less than or equal to inverted right perpendicular (n - 2)/2 inverted left perpendicular. (C) 1997 Academic Press.