Improved Euclidean Shallow Light Trees

2026-08-04

AI Summary

This work addresses the construction of shallow-light trees (SLTs) in the Euclidean plane that simultaneously achieve near-shortest root-to-vertex paths—specifically, $(1+\varepsilon)$-root-stretch—and low lightness. Tackling the long-standing bottleneck of the $2/\varepsilon$ lightness lower bound barrier, the paper introduces a novel approach grounded in geometric graph theory and the structural properties of Euclidean minimum spanning trees, combined with refined tree-building and edge-replacement strategies. The authors significantly improve the best-known upper bound on SLT lightness from $2/\varepsilon$ to $(5/3 + o_\varepsilon(1))/\varepsilon$, and further optimize it to approximately $(0.987 + o_\varepsilon(1))/\varepsilon$. This result closely approaches the current theoretical lower bound of $0.862/\varepsilon$, substantially surpassing classical constructions.

Abstract

For parameters $α,β\geq 1$, a spanning tree $T$ of a weighted graph $G$ rooted at a designated vertex $r$ is called an $(α,β)$-shallow-light tree (SLT) if (i) for every vertex $v$, $d_T(r,v) \leq α\cdot d_G(r,v)$ (root-stretch $α$), and (ii) $w(T) \leq β\cdot w(\mathsf{MST})$ (lightness $β$). The pioneering work of Khuller, Raghavachari, and Young (SODA 1993) constructed $\left(1+ε, \tfrac{2}ε+1\right)$-SLTs for general weighted graphs, and proved that this tradeoff between root-stretch and lightness is tight even for series-parallel graphs. They further asked whether even a slight improvement, namely reducing the lightness to $\tfrac{2-c}ε$ for any constant $c>0$, is possible in the Euclidean plane. We resolve this longstanding question in the affirmative. Specifically, we show that every Euclidean instance admits an SLT with root-stretch $1+ε$ and lightness at most $\left(\frac{5}{3} + o_ε(1)\right) \cdot \frac{1}ε$, thereby significantly improving upon the longstanding $2/ε$ barrier. As our second main result, we provide a construction of SLTs in the Euclidean plane, with root stretch $1+ε$ and lightness at most $\left(\frac{2π}{\sqrt{4π^2+1}}+o_ε(1)\right)\frac{1}ε \approx (0.987+o_ε(1))\frac{1}ε$. Notably, this reduces the leading $2/ε$ term in the lightness bound by more than a factor of two, and comes quite close to the lower bound of $\left(\frac{2π}{2π+1} +o_ε(1))\right) \cdot \frac{1}ε \approx (0.862 +o_ε(1))\frac{1}ε$ by Elkin and Solomon (FOCS 2011).

Problem

• Shallow-Light Tree
• Euclidean plane
• root-stretch
• lightness
• spanning tree

Innovation

• Shallow-Light Tree
• Euclidean plane
• lightness
• root-stretch
• spanning tree