On the Restricted $k$-Steiner Tree Problem
Year: 2023
Type: preprint
Abstract: Given a set $P$ of $n$ points in $\mathbb{R}^2$ and an input line $γ$ in $\mathbb{R}^2$, we present an algorithm that runs in optimal $Θ(n\log n)$ time and $Θ(n)$ 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 $s \in γ$, 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. ("Generalised k-Steiner Tree Problems in Normed Planes", arXiv:1111.1464) 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. ("Generalised k-Steiner Tree Problems in Normed Planes", arXiv:1111.1464) 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.
Source: arXiv (Cornell University)
Authors: Bose, Prosenjit, D'Angelo, Anthony, Durocher, Stephane
Institutions: Carleton University, University of Manitoba
Language: English
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6891935828
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