Greedy Completion for Weighted $(\alpha,\beta)$-Spanners

TL;DR

This paper introduces a greedy completion method for constructing weighted graph spanners with improved size bounds, generalizing previous additive completion techniques and achieving new results for specific spanner parameters.

Contribution

It presents a simple greedy completion procedure for weighted graph spanners, leading to the first known $(k,k-1)$-spanners with near-linear size.

Findings

Constructed $(k,k-1)$-spanners with size $ ilde{O}(n^{1+1/k})$

Generalized additive completion to weighted graphs

Achieved new bounds for spanner sizes in weighted graphs

Abstract

We study $(\alpha,\beta)$-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.