The Steiner tree problem in orientation metrics

Yan, G. Y., Albrecht, Andreas A., Young, G. H. F. and Wong, C. K. (1997) The Steiner tree problem in orientation metrics. Journal of Computer and System Sciences, 55 (3) . pp. 529-546. ISSN 0022-0000 [Article] (doi:10.1006/jcss.1997.1513)

Abstract

Given a setΘofαi(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 areσorientations, forming anglesiπ/σ, 0⩽i⩽σ−1, with thex-axis, whereσ⩾2 is an integer, we call the metric aλσ-metric. Note that theλ2-metric is the well-known rectilinear metric and theλ∞corresponds to the Euclidean metric. In this paper, we will concentrate on theλ3-metric. In theλ2-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 directionsiπ/2,i=0, 1, through all demand points. But this is not true in theλ3-metric. In this paper, we mainly prove the following theorem: LetP,Q, andOi(i=1, 2, …, k) be the set ofndemand points, the set of Steiner points, and the set of theith generation intersection points, respectively. Then there exists a solutionGof the Steiner problemSnsuch that all Steiner points are in ∪ki=1 Oi, wherek⩽⌈ (n−2)/2⌉.

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