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 ε > 0, our algorithm runs in space O^e(klog(N · wmax/wmin)) where wmax and wmin are the maximum and minimum edge weights in G, and produces a weighted graph H with O^e(n^1+2/k/ε²) edges that spectrally approximates G, in the sense of Spielmen and Teng [52], 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 [51] and uses results from recent work on space-bounded Laplacian solvers [41]. In particular, we demonstrate an inherent tradeoff (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.