The bottleneck 2-connected $k$-Steiner network problem for $k\leq 2$

TL;DR

This paper develops efficient algorithms for constructing 2-connected Euclidean Steiner networks with minimal bottleneck edge length for up to two added points, extending to various norms.

Contribution

It introduces exact algorithms with quadratic and near-quadratic complexity for the 2-connected bottleneck Steiner network problem for k≤2, utilizing geometric graph structures.

Findings

Algorithms run in O(n^2) for k=1 and O(n^2 log n) for k=2.

Methods extend to other normed planes like L_p.

Solutions ensure 2-connected networks with minimized longest edge.

Abstract

The geometric bottleneck Steiner network problem on a set of vertices $X$ embedded in a normed plane requires one to construct a graph $G$ spanning $X$ and a variable set of $k\geq 0$ additional points, such that the length of the longest edge is minimised. If no other constraints are placed on $G$ then a solution always exists which is a tree. In this paper we consider the Euclidean bottleneck Steiner network problem for $k\leq 2$, where $G$ is constrained to be 2-connected. By taking advantage of relative neighbourhood graphs, Voronoi diagrams, and the tree structure of block cut-vertex decompositions of graphs, we produce exact algorithms of complexity $O(n^2)$ and $O(n^2\log n)$ for the cases $k=1$ and $k=2$ respectively. Our algorithms can also be extended to other norms such as the $L_p$ planes.