🤖 AI Summary
This study addresses the construction of Steiner shallow-light trees (SLTs) in high-dimensional Euclidean spaces, aiming to simultaneously achieve near-shortest paths and low lightness—both independent of the ambient dimension. For any finite point set, designated root, and accuracy parameter ε, the authors propose a method grounded in geometric graph theory and approximation algorithms. By introducing Steiner points and reanalyzing the planar core structure along with its high-dimensional generalization, they construct an SLT with stretch factor (1+ε) and lightness O(√(1/ε)). This result provides the first dimension-independent performance guarantee for SLTs in arbitrary dimensions, thereby affirming Solomon’s conjecture on the existence of such structures in high-dimensional Euclidean spaces.
📝 Abstract
This paper proves a conjecture by Solomon about Steiner shallow-light trees (SLT) in Euclidean $d$-space: It is shown that for any finite point set $\mathbb{R}^d$, any root, and any $ε>0$, there is a Euclidean Steiner $(1+ε,O(\sqrt{1/ε}))$-SLT without any dependence on dimension. We also revisit the core example, designed by Solomon, in the plane and its generalization to $d$-space.