Methods to Construct Shortest Trees

Abstract

In the chapters before, we described the combinatorial structure of shortest trees; and we investigated a local solution to determine the locus of the Steiner points in relation to its neighbors. The purpose of this chapter is to create methods which find solutions of Steiner’s Problem and its relatives in general. It will not be possible to develop an exact solution method working in all Banach-Minkowski spaces or at least in all planes.¹ At first we have to define what “solve” means. We will find several strategies to solve Steiner’s Problem in specific spaces. Moreover, we will describe methods to construct shortest trees in classes of Banach-Minkowski planes and in some higher dimensional spaces, too. On the other hand, we will investigate heuristics and approximations.