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The Euclidean Bottleneck Full Steiner Tree Problem

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Abstract

Given two sets in the plane, R of n (terminal) points and S of m (Steiner) points, a full Steiner tree is a Steiner tree in which all points of R are leaves. In the bottleneck full Steiner tree (BFST) problem, one has to find a full Steiner tree T (with any number of Steiner points from S), such that the length of the longest edge in T is minimized, and, in the k-BFST problem, has to find a full Steiner tree T with at most km Steiner points from S such that the length of the longest edge in T is minimized. The problems are motivated by wireless network design.

In this paper, we present an exact algorithm of \({{{\mathcal {O}}}}((n+m)\log^{2}{m})\) time to solve the BFST problem. Moreover, we show that the k-BFST problem is NP-hard and that there exists a polynomial-time approximation algorithm for the problem with performance ratio 4.

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Acknowledgements

The author would like to thank Matthew Katz and Paz Carmi for helpful discussions.

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Correspondence to A. Karim Abu-Affash.

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Abu-Affash, A.K. The Euclidean Bottleneck Full Steiner Tree Problem. Algorithmica 71, 139–151 (2015). https://doi.org/10.1007/s00453-013-9788-x

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