Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

Abstract

We present an almost-linear time algorithm for constructing a spectral sparsifier with the number of edges linear in its number of vertices. This improves all previous constructions of linear-sized spectral sparsifiers, which require $\Omega(n^2)$ time. A key ingredient in our algorithm is a novel combination of two techniques used in literature for constructing spectral sparsifiers: random sampling by effective resistance, and adaptive construction based on barrier functions.