Given a set P of n points in R2\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^2$$\end{document} and an input line γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} in R2\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^2$$\end{document}, we present an algorithm that runs in optimal Θ(nlogn)\documentclass[12pt]{minimal}
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\begin{document}$$\varTheta (n\log n)$$\end{document} time and Θ(n)\documentclass[12pt]{minimal}
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\begin{document}$$\varTheta (n)$$\end{document} space to solve a restricted version of the 1-Steiner tree problem. Our algorithm returns a minimum-weight tree interconnecting P using at most one Steiner point s∈γ\documentclass[12pt]{minimal}
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\begin{document}$$s \in \gamma $$\end{document}, where edges are weighted by the Euclidean distance between their endpoints. We then extend the result to j input lines. Following this, we show how the algorithm of Brazil et al. in Algorithmica 71(1):66–86 that solves the k-Steiner tree problem in R2\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^2$$\end{document} in O(n2k)\documentclass[12pt]{minimal}
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\begin{document}$$O(n^{2k})$$\end{document} time can be adapted to our setting. For k>1\documentclass[12pt]{minimal}
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\begin{document}$$k>1$$\end{document}, restricting the (at most) k Steiner points to lie on an input line, the runtime becomes O(nk)\documentclass[12pt]{minimal}
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\begin{document}$$O(n^{k})$$\end{document}. Next we show how the results of Brazil et al. in Algorithmica 71(1):66–86 allow us to maintain the same time and space bounds while extending to some non-Euclidean norms and different tree cost functions. Lastly, we extend the result to j input curves.
