Euclidean Steiner trees optimal with respect to swapping 4-point subtrees

Abstract

The Steiner tree problem in Euclidean space \(E^3\) asks for a minimum length network \(T\), called a Euclidean Steiner Minimum Tree (ESMT), spanning a given set of points. This problem is NP-hard and the hardness is inherently due to the number of feasible topologies (underlying graph structure of \(T\)) which increases exponentially as the number of given points increases. Planarity is a very strong condition that gives a big difference between the ESMT problem in the Euclidean plane \(E^2\) and in Euclidean \(d\)-space \(E^d (d\ge 3)\): the ESMT problem in the plane is practically solvable whereas the ESMT problem in \(d\)-space is really intractable. The simplest tree rearrangement technique is to repeatedly replace a subtree spanning four nodes in \(T\) with another subtree spanning the same four nodes. This technique is referred to as the Swapping 4-point Topology/ Tree technique in this paper. An indicator (or quasi-indicator) of \(T\) plays a similar role in the optimization of the length \(L(T)\) of \(T\) in the discrete topology space (the underlying graph structure of \(T\)) to the derivative of a differentiable function which indicates a fastest direction of descent. \(T\) will be called S4pT-optimal if it is optimal with respect to swapping 4-point subtrees. In this paper we first make a complete analysis of 4-point trees in Euclidean space exploring all possible types of 4-point trees and their properties. We review some known indicators of 4-point ESMTs in \(E^2\), and give some simple geometric proofs of these indicators. Then, we translate these indicators to \(E^3\), producing eight quasi-indicators in \(E^3\) using computational experiments, the best quasi-indicator \(\rho _\mathrm{osr}\) is sifted with an effectiveness for randomly generated 4-point sets as high as 98.62 %. Finally we show how \(\rho _\mathrm{osr}\) is used to find an S4pT-optimal ESMT on 14 probability vectors in \(4d\)-space with a detailed example.