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 <= log n, and an error parameter epsilon > 0, our algorithm runs in space (O) over tilde (k log(N center dot w(max)/w(min))), where wmax and wmin are the maximum and minimum edge weights in G, and produces a weighted graph H with (O) over tilde (n(1+2/k)/epsilon(2)) edges that spectrally approximates G, in the sense of Spielman and Teng, up to an error of epsilon. 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.