🤖 AI Summary
This paper studies the construction of lightweight sparse distance-preserving subgraphs (light spanners) of graphs. For the weight analysis of greedy spanners, it introduces the first method that directly applies Moore bound reasoning to lightness proofs—integrating hierarchical clustering, edge-weight normalization, and Moore-type counting arguments into a concise, general, and direct analytical framework. The authors rigorously prove that the greedy spanner with stretch $(1+varepsilon)(2k-1)$ achieves lightness $O_varepsilon(n^{1/k})$, where the $varepsilon$-dependence is $O(1/varepsilon)$—improving upon all prior non-greedy constructions and yielding the best $varepsilon$-sensitivity improvement since STOC’23. Moreover, this framework unifies and significantly simplifies the analyses of classical results by Chechik–Wulff-Nilsen and Le–Solomon.
📝 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.