An exact algorithm for the bottleneck 2-connected $k$-Steiner network problem in $L_p$ planes

TL;DR

This paper introduces the first exact polynomial-time algorithm for constructing optimal 2-connected Steiner networks with at most k Steiner points in Lp planes, minimizing the longest edge for energy-efficient wireless networks.

Contribution

It provides a novel polynomial-time algorithm for a geometric network design problem with Steiner points, using Voronoi diagrams and graph decompositions.

Findings

Algorithm runs in O(n^k log^{5k/2} n) for 1<p<∞

Algorithm runs in O(n^2 log^{(7k/2)+1} n) for p in {1, ∞}

First exact polynomial-time solution for this class of problems.

Abstract

We present the first exact polynomial time algorithm for constructing optimal geometric bottleneck 2-connected Steiner networks containing at most $k$ Steiner points, where $k>2$ is a constant. Given a set of $n$ vertices embedded in an $L_p$ plane, the objective of the problem is to find a 2-connected network, spanning the given vertices and at most $k$ additional vertices, such that the length of the longest edge is minimised. In contrast to the discrete version of this problem the additional vertices may be located anywhere in the plane. The problem is motivated by the modelling of relay-augmentation for the optimisation of energy consumption in wireless ad hoc networks. Our algorithm employs Voronoi diagrams and properties of block-cut-vertex decompositions of graphs to find an optimal solution in $O(n^k\log^{\frac{5k}{2}}n)$ steps when $1<p<\infty$ and in $O(n^2\log^{\frac{7k}{2}+1}n)$ steps when $p\in\{1,\infty\}$.