Analysis of Kelner and Levin graph sparsification algorithm for a streaming setting

TL;DR

This paper provides a new proof that confirms the Kelner and Levin graph sparsification algorithm effectively produces spectral sparsifiers in streaming settings, addressing previous analytical flaws.

Contribution

It offers a rigorous, dependency-aware proof of the algorithm's correctness, improving upon the original analysis by fixing a flaw.

Findings

Proves the algorithm produces spectral sparsifiers with high probability

Addresses dependencies across resparsifications using martingale inequalities

Fixes a flaw in the original proof

Abstract

We derive a new proof to show that the incremental resparsification algorithm proposed by Kelner and Levin (2013) produces a spectral sparsifier in high probability. We rigorously take into account the dependencies across subsequent resparsifications using martingale inequalities, fixing a flaw in the original analysis.