A Framework for Analyzing Resparsification Algorithms

A spectral sparsifier of a graph G is a sparser graph H that approximately preserves the quadratic form of G, i.e. for all vectors x, xLGx ≈ x T LHx, where LG and LH denote the respective graph Laplacians. Spectral sparsifiers generalize cut sparsifiers, and have found many applications in designing graph algorithms. In recent years, there has been interest in computing spectral sparsifiers in semi-streaming and dynamic settings. Natural algorithms in these settings often involve repeated sparsification of a graph, and in turn accumulation of errors across these steps. We present a framework for analyzing algorithms that perform repeated sparsifications that only incur error corresponding to a single sparsification step, leading to better results for many of these resparsification-based algorithms. As an application, we show how to maintain a spectral sparsifier in the semi-streaming setting: We present a simple algorithm that, for a graph G on n vertices and m edges, computes a spectral sparsifier of G with O(n log n) edges in a single pass over G, using only O(n log n) space, and O(m log n) total time. This improves on previous best semi-streaming algorithms for both spectral and cut sparsifiers by a factor of log n in both space and runtime. The algorithm extends to semi-streaming row sampling for general PSD matrices. We also use our framework to combine a spectral sparsification algorithm by Koutis with improved spanner constructions to give a parallel algorithm for constructing O(n log n log log n) sized spectral sparsifiers in O(m log n log log n) time. This is the best combinatorial graph sparsification algorithm to date, and the size of the sparsifiers produced is only a factor log n log log n more than ones produced by numerical routines.