Approximating Minimum Steiner Point Trees in Minkowski Planes

TL;DR

This paper investigates approximating minimum Steiner point trees in Minkowski planes, showing bounds on performance differences and introducing a canonical form for these trees, with implications for Euclidean and Minkowski geometries.

Contribution

It proposes using Steiner minimal trees as approximations, establishes bounds on their performance difference in various metric spaces, and introduces a canonical form for minimum Steiner point trees in the Euclidean plane.

Findings

Performance difference at most 2n-4 in arbitrary metric spaces

Optimality of the bound in Euclidean plane

New canonical form for minimum Steiner point trees in Euclidean plane

Abstract

Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number of additional points is minimized. We propose using Steiner minimal trees to approximate minimum Steiner point trees. It is shown that in arbitrary metric spaces this gives a performance difference of at most $2n-4$, where $n$ is the number of terminals. We show that this difference is best possible in the Euclidean plane, but not in Minkowski planes with parallelogram unit balls. We also introduce a new canonical form for minimum Steiner point trees in the Euclidean plane; this demonstrates that minimum Steiner point trees are shortest total length trees with a certain discrete-edge-length condition.