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An Alternate Proof of Near-Optimal Light Spanners

Greg Bodwin

TL;DR

The paper presents a direct, Moore-bounds–style analysis of the greedy spanner to obtain near-optimal light spanners with improved $\varepsilon$-dependence. By reducing to graphs with weighted girth and unit-weight spanning cycles, it develops a bucket-monotone path framework that tightens the counting bounds and avoids previous $k$-factor losses. The core contributions are the weighted-girth reductions, bucket-based path safety notions, and a robust counting lemmas suite (weak/medium/full) built around bucket-monotone and hiker-path constructions. Together, these yield a proof of the lightness bound matching the state of the art up to $\varepsilon$-dependence and extend the Moore-bound intuition to light-spanner sparsity.

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.

An Alternate Proof of Near-Optimal Light Spanners

TL;DR

The paper presents a direct, Moore-bounds–style analysis of the greedy spanner to obtain near-optimal light spanners with improved -dependence. By reducing to graphs with weighted girth and unit-weight spanning cycles, it develops a bucket-monotone path framework that tightens the counting bounds and avoids previous -factor losses. The core contributions are the weighted-girth reductions, bucket-based path safety notions, and a robust counting lemmas suite (weak/medium/full) built around bucket-monotone and hiker-path constructions. Together, these yield a proof of the lightness bound matching the state of the art up to -dependence and extend the Moore-bound intuition to light-spanner sparsity.

Abstract

In 2016, a breakthrough result of Chechik and Wulff-Nilsen [SODA '16] established that every -node graph has a -spanner of lightness , and recent followup work by Le and Solomon [STOC '23] generalized the proof strategy and improved the dependence on . We give a new proof of this result, with the improved -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.
Paper Structure (18 sections, 20 theorems, 11 equations, 2 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 20 theorems, 11 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Other Related Work
  3. Organization
  4. Warmup 1: The Moore Bounds for Spanner Sparsity
  5. Lightness Reductions from Prior Work
  6. The Weighted Girth Framework
  7. Reduction to Unit-Weight Spanning Cycles
  8. Warmup 2: Lightness Bounds via Monotone Paths
  9. Edge-Safe Paths and Warmup Proof Intuition
  10. Monotone Safe Paths and the Dispersion Lemma
  11. Hiker Paths and the Counting Lemma
  12. Full Proof: Light Spanners via Bucket-Monotone Paths
  13. Bucket-Safe Paths and Full Proof Intuition
  14. Special Case 1: Well-Separated Edge Weights.
  15. Special Case 2: Bucketed Edge Weights.
  16. ...and 3 more sections

Key Result

Theorem 1.2

For all positive integers $k, n$, every $n$-node graph $G$ has a $(2k-1)$-spanner $H$ on $|E(H)| \le O(n^{1+1/k})$ edges. This tradeoff is best possible, assuming the girth conjecture girth.

Figures (2)

  • Figure 5: A path that is safe for the edge $(u, v)$ (considering the counterclockwise direction around the spanning cycle to forward, and clockwise to be backward)
  • Figure 6: A counterexample to the dispersion lemma for safe $k$-paths

Theorems & Definitions (32)

  • definition 1.1: Spanners PU89jacmPU89sicomp
  • Theorem 1.2: ADDJS93
  • definition 1.3: Spanner Lightness
  • Theorem 1.4: CW18
  • Theorem 2.1: Moore Bounds
  • Lemma 2.2: Unweighted Dispersion Lemma
  • Lemma 2.3: Unweighted Weak Counting Lemma
  • Lemma 2.4: Unweighted Medium Counting Lemma
  • Lemma 2.5: Unweighted Full Counting Lemma
  • definition 3.1: Normalized Weight and Weighted Girth ENS15
  • ...and 22 more