On Improved Bounds on Bounded Degree Spanning Trees for Points in Arbitrary Dimension

TL;DR

This paper establishes improved bounds for the weight of degree-3 spanning trees in Euclidean spaces of arbitrary dimension, providing both upper and lower bounds and a linear-time construction method.

Contribution

It proves new upper and lower bounds on the weight ratio of degree-3 spanning trees relative to minimum spanning trees in arbitrary dimensions.

Findings

Existence of degree-3 spanning trees within 1.559 times the MST weight

Construction method for such trees in linear time

Lower bound example showing ratio cannot be less than 1.447

Abstract

Given points in Euclidean space of arbitrary dimension, we prove that there exists a spanning tree having no vertices of degree greater than 3 with weight at most 1.559 times the weight of the minimum spanning tree. We also prove that there is a set of points such that no spanning tree of maximal degree 3 exists that has this ratio be less than 1.447. Our central result is based on the proof of the following claim: Given $n$ points in Euclidean space with one special point $V$, there exists a Hamiltonian path with an endpoint at $V$ that is at most 1.559 times longer than the sum of the distances of the points to $V$. These proofs also lead to a way to find the tree in linear time given the minimal spanning tree.