On the existence of shortest directed networks

TL;DR

This paper proves that shortest directed networks connecting two finite sets exist in certain metric spaces and provides bounds on the number of Steiner points, extending known results beyond the Euclidean plane.

Contribution

It establishes the existence of shortest directed networks in finitely compact metric spaces with geodesics, generalizing previous Euclidean plane results and bounding Steiner points.

Findings

Existence of shortest directed networks in general metric spaces.

Bound on the number of Steiner points based on set sizes.

Extension of Euclidean plane results to broader metric spaces.

Abstract

A directed network connecting a set A to a set B is a digraph containing an a-b path for each a in A and b in B. Vertices in the directed network not in A or B are called Steiner points. We show that in a finitely compact metric space in which geodesics exist, any two finite sets A and B are connected by a shortest directed network. We also bound the number of Steiner points by a function of the sizes of A and B. Previously, such an existence result was known only for the Euclidean plane [M. Alfaro, Pacific J. Math. 167 (1995) 201-214]. The main difficulty is that, unlike the undirected case (Steiner minimal trees), the underlying graphs need not be acyclic. Existence in the undirected case was first shown by E. J. Cockayne [Canad. Math. Bull. 10 (1967) 431-450].