Given a set P of n points in \(\mathbb {R}^2\) and an input line \(\gamma \) in \(\mathbb {R}^2\), we present an algorithm that runs in optimal \(\varTheta (n\log n)\) time and \(\varTheta (n)\) 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 \(s \in \gamma \), 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 \(\mathbb {R}^2\) in \(O(n^{2k})\) time can be adapted to our setting. For \(k>1\), restricting the (at most) k Steiner points to lie on an input line, the runtime becomes \(O(n^{k})\). 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.

给定一组P的Ñ在点\（\ mathbb {R} ^ 2 \）和输入线\（\伽马\）在\（\ mathbb {R} ^ 2 \），我们提出了一种算法，在最优运行\(\varTheta (n\log n)\)时间和\(\varTheta (n)\)空间来解决 1-Steiner 树问题的受限版本。我们的算法返回一个最小权重树，最多使用一个 Steiner 点\(s \in \gamma \)互连P，其中边由端点之间的欧几里得距离加权。然后我们将结果扩展到j输入线。在此之后，我们展示了巴西等人的算法如何。在 Algorithmica 71(1):66–86 中解决了\(\mathbb {R}^2\)中\(O(n^{2k})\)时间的k -Steiner 树问题可以适应我们的设置. 对于\(k>1\)，限制（最多）k 个Steiner 点位于输入线上，运行时间变为\(O(n^{k})\)。接下来，我们展示了巴西等人的结果如何。在 Algorithmica 71(1):66-86 中允许我们保持相同的时间和空间界限，同时扩展到一些非欧式范数和不同的树成本函数。最后，我们将结果扩展到j 个输入曲线。