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2009•Discrete Mathematics Algorithms and ApplicationsRequires access

IDENTIFYING STEINER MINIMAL TREES ON FOUR POINTS IN SPACE

Jia Feng Weng, Doreen Anne Thomas, Iven Mareels

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Abstract

A Steiner minimal tree is a network with minimum length spanning a given set of points in space. There are several criteria for identifying the Steiner minimal tree on four points in the Euclidean plane. However, it has been proved that the length of the Steiner minimal tree on four points cannot be computed using radicals if the four points lie in Euclidean space. This unsolvability implies that it is unlikely that similar necessary and sufficient conditions exist in the spatial case. Hence, a problem arises: Is it possible to generalize the known planar criteria to space in the sense that they are sufficient to identify Steiner minimal trees on four points in space? This problem is investigated and some sufficient conditions are proved in this paper. These sufficient conditions can help us to solve the general Steiner tree problem on n(> 4) points in Euclidean space.

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What this paper is about

A Steiner minimal tree is a network with minimum length spanning a given set of points in space. There are several criteria for identifying the Steiner minimal tree on four points in the Euclidean plane. However, it has been proved that the length of the Steiner minimal tree on four points cannot be computed using radicals if the four points lie in Euclidean space. This unsolvability implies that it is unlikely that similar necessary and sufficient conditions exist in the spatial case. Hence, a problem arises: Is it possible to generalize the known planar criteria to space in the sense that they are sufficient to identify Steiner minimal trees on four points in space? This problem is investigated and some sufficient conditions are proved in this paper. These sufficient conditions can help us to solve the general Steiner tree problem on n(> 4) points in Euclidean space.

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Available abstract

A Steiner minimal tree is a network with minimum length spanning a given set of points in space. There are several criteria for identifying the Steiner minimal tree on four points in the Euclidean plane. However, it has been proved that the length of the Steiner minimal tree on four points cannot be computed using radicals if the four points lie in Euclidean space. This unsolvability implies that it is unlikely that similar necessary and sufficient conditions exist in the spatial case. Hence, a problem arises: Is it possible to generalize the known planar criteria to space in the sense that they are sufficient to identify Steiner minimal trees on four points in space? This problem is investigated and some sufficient conditions are proved in this paper. These sufficient conditions can help us to solve the general Steiner tree problem on n(> 4) points in Euclidean space.

Key concepts: Steiner tree problem, Mathematics, Combinatorics, Euclidean space, Space (punctuation), Euclidean geometry, Minimum spanning tree, Spanning tree

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