# Generalised k-Steiner tree problems in normed planes

Brazil, Marcus, Ras, Charl J., Swanepoel, Konrad and Thomas, Doreen A. (2015) Generalised k-Steiner tree problems in normed planes. Algorithmica , 71 (1). pp. 66-86. ISSN 1432-0541

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## Abstract

The 1-Steiner tree problem, the problem of constructing a Steiner minimum tree containing at most one Steiner point, has been solved in the Euclidean plane by Georgakopoulos and Papadimitriou using plane subdivisions called oriented Dirichlet cell partitions. Their algorithm produces an optimal solution within $\mathcal{O}(n^2)$ time. In this paper we generalise their approach in order to solve the $k$-Steiner tree problem, in which the Steiner minimum tree may contain up to $k$ Steiner points for a given constant $k$. We also extend their approach further to encompass arbitrary normed planes, and to solve a much wider class of problems, including the $k$-bottleneck Steiner tree problem and other generalised $k$-Steiner tree problems. We show that, for any fixed $k$, such problems can be solved in $\mathcal{O}(n^{2k})$ time.

Item Type: Article http://link.springer.com/journal/453 © 2015 Springer Science+Business Media New York Q Science > QA Mathematics Departments > Mathematics ARC Discovery Grant 23 Jan 2012 16:11 http://eprints.lse.ac.uk/41657/