On the Restricted 1-Steiner Tree Problem

Abstract

Given a set P of n points in R 2 and an input line γ in 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 ∈ 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.[10] that solves the k-Steiner tree problem in 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. [10] 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.

Proposing to improve the Heuristic Algorithms to Solve a Steiner-minimal-tree Problem in Large Size Sparse Graphs

“…Due to its scientific nature and wide applicability, SMT problems have attracted the attention of many scientists in the world for decades [27,35,44]. Some of them are Martin Zachariasen [21], Tobias Polzin [37], Pieter Oloff De Wet [26], Xinhui Wang [40], Jon William Van Laarhoven [17], and Zhiliu Zhang [45].…”

Constrained Geometric Optimization Problems

“…Details appear in Chapter 2. A preliminary version of this result appeared in COCOON 2020 under the title "On the restricted 1-Steiner tree problem" [21], and the full version of this result appeared in the Journal of Combinatorial Optimization in 2021 under the title "On the restricted k-Steiner tree problem" [22].…”