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.