Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

TL;DR

This paper introduces an almost-linear time algorithm for creating spectral sparsifiers of graphs with linear size, significantly improving the efficiency over previous methods that required quadratic time.

Contribution

It presents the first nearly linear time algorithm for linear-sized spectral sparsification, combining random sampling and barrier function techniques.

Findings

Achieves spectral sparsification in almost-linear time

Reduces complexity from a0a0n^2 to nearly linear

Combines two advanced techniques for improved efficiency

Abstract

We present the first almost-linear time algorithm for constructing linear-sized spectral sparsification for graphs. This improves all previous constructions of linear-sized spectral sparsification, which requires $\Omega(n^2)$ time. A key ingredient in our algorithm is a novel combination of two techniques used in literature for constructing spectral sparsification: Random sampling by effective resistance, and adaptive constructions based on barrier functions.