Approximating Minimum Steiner Point Trees in Minkowski Planes

M. Brazil; C. J. Ras; D. A. Thomas

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arXiv · 1307.2987

Approximating Minimum Steiner Point Trees in Minkowski Planes

M. Brazil·C. J. Ras·D. A. Thomas

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Abstract

Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number of additional points is minimized. We propose using Steiner minimal trees to approximate minimum Steiner point trees. It is shown that in arbitrary metric spaces this gives a performance difference of at most $2n-4$, where $n$ is the number of terminals. We show that this difference is best possible in the Euclidean plane, but not in Minkowski planes with parallelogram unit balls. We also introduce a new canonical form for minimum Steiner point trees in the Euclidean plane; this demonstrates that minimum Steiner point trees are shortest total length trees with a certain discrete-edge-length condition.

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BibTeX RIS

M. Brazil, C. J. Ras, D. A. Thomas. 2013-07-11. Approximating Minimum Steiner Point Trees in Minkowski Planes. https://arxiv.org/abs/1307.2987

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