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Greedy Completion for Weighted $(α,β)$-Spanners

Elad Tzalik

Abstract

We study $(α,β)$-spanners for weighted graphs. We propose a simple greedy completion procedure which starts from a sparse initial graph, and repeatedly fixes pairs of vertices with a bad stretch, generalizing Kunedsen's additive completion [SWAT '14]. As an application, we construct $(k,k-1)$-spanners for weighted graphs of size $\tilde{O}(n^{1+1/k})$, which were previously unknown.

Greedy Completion for Weighted $(α,β)$-Spanners

Abstract

We study -spanners for weighted graphs. We propose a simple greedy completion procedure which starts from a sparse initial graph, and repeatedly fixes pairs of vertices with a bad stretch, generalizing Kunedsen's additive completion [SWAT '14]. As an application, we construct -spanners for weighted graphs of size , which were previously unknown.
Paper Structure (25 sections, 9 theorems, 24 equations)

This paper contains 25 sections, 9 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. The quest to handle weights.
  3. Handling weights: beyond additive stretch.
  4. The "local" recipe for $(\alpha,\beta)$ spanners, and how it fails to handle weights.
  5. Conceptual contribution.
  6. Comparison to PopovaTzalikITCS.
  7. Acknowledgments
  8. Technical Overview
  9. Knudsen's algorithm
  10. Defining the $(\alpha,\beta)$-completion
  11. Attempt 1 (unsuccessful): naive $(\alpha,\beta)$-completion
  12. Attempt 2 (unsuccessful): $(\alpha,\beta)$-completion by fixing the bottlenecks
  13. Attempt 3 (successful): incorporating shortcuts
  14. The Baswana-Sen algorithm, and its properties
  15. The $(k,k-1)$-spanner algorithm, and its analysis
  16. ...and 10 more sections

Key Result

Theorem 1.1

For $k\geq 2$ and any weighted graph $G$, there exists a $(k,k-1)$ spanner $H$ of $G$ of size $\Tilde{O}(n^{1+1/k})$. Moreover, $H$ can be computed in polynomial time.

Theorems & Definitions (31)

  • Definition 1: $(\alpha,\beta)$-spanner
  • Theorem 1.1
  • Theorem 2.1: baswana2007simple
  • Remark 1
  • Definition 2: $\alpha$-segmentation
  • Definition 3: minimal segmentation
  • Definition 4: Distant edges of a segmentation
  • Lemma 1
  • proof
  • Corollary 1
  • ...and 21 more