# 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
(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⌉.

Item Type: | Article |
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Research Areas: | A. > School of Science and Technology > Computer Science |

Item ID: | 12412 |

Depositing User: | Andreas Albrecht |

Date Deposited: | 12 Nov 2013 08:26 |

Last Modified: | 12 Jun 2019 12:45 |

URI: | https://eprints.mdx.ac.uk/id/eprint/12412 |

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