Small-Space Spectral Sparsification via Bounded-Independence Sampling

We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph G on n vertices described by a binary string of length N, an integer k ≤ logn, and an error parameter ϵ > 0, our algorithm runs in space Õ(k log(N · w_max/w_min)), where w_max and w_min are the maximum and minimum edge weights in G, and produces a weighted graph H with Õ(n^1+2/k/ϵ²) edges that spectrally approximates G, in the sense of Spielman and Teng, up to an error of ϵ. Our algorithm is based on a new bounded-independence analysis of Spielman and Srivastava's effective resistance-based edge sampling algorithm and uses results from recent work on space-bounded Laplacian solvers. In particular, we demonstrate an inherent trade-off (via upper and lower bounds) between the amount of (bounded) independence used in the edge sampling algorithm, denoted by k above, and the resulting sparsity that can be achieved.