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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A new bound for the Steiner ratio
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by D. Z. Du and F. K. Hwang PDF
Trans. Amer. Math. Soc. 278 (1983), 137-148 Request permission

Abstract:

Let $V$ denote a given set of $n$ points in the euclidean plane. A Steiner minimal tree for $V$ is the shortest network (clearly, it has to be a tree) interconnecting $V$. Junctions of the network which are not in $V$ are called Steiner points (those in $V$ will be called regular points). A shortest tree interconnecting $V$ without using any Steiner points is called a minimal tree. Let $\sigma (V)$ and $\mu (V)$ denote the lengths of a Steiner minimal tree and a minimal tree, respectively. Define $\rho$ to be the greatest lower bound for the ratio $\sigma (V)/\mu (V)$ over all $V$. We prove $\rho > .8$.
References
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 278 (1983), 137-148
  • MSC: Primary 05C05; Secondary 51M15
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0697065-3
  • MathSciNet review: 697065