An Alternate Proof of Near-Optimal Light Spanners

TL;DR

This paper presents a new proof for the existence of light spanners in graphs, improving the dependence on epsilon by analyzing the greedy spanner directly, extending classical Moore bounds.

Contribution

It provides an alternative proof of a key light spanner result with better epsilon dependence, using a direct analysis of the greedy spanner.

Findings

Improved epsilon dependence in light spanner construction

Direct analysis of the greedy spanner approach

Extension of Moore bounds to spanner analysis

Abstract

In 2016, a breakthrough result of Chechik and Wulff-Nilsen [SODA '16] established that every $n$-node graph $G$ has a $(1+\varepsilon)(2k-1)$-spanner of lightness $O_{\varepsilon}(n^{1/k})$, and recent followup work by Le and Solomon [STOC '23] generalized the proof strategy and improved the dependence on $\varepsilon$. We give a new proof of this result, with the improved $\varepsilon$-dependence. Our proof is a direct analysis of the often-studied greedy spanner, and can be viewed as an extension of the folklore Moore bounds used to analyze spanner sparsity.