Small-Space Spectral Sparsification via Bounded-Independence Sampling

Abstract

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 \(\tilde{O}(k\log (N\cdot w_{\mathrm{max}}/w_{\mathrm{min}})) \) where w max and w min are the maximum and minimum edge weights in G , and produces a weighted graph H with \(\tilde{O}(n^{1+2/k}/\epsilon ^2) \) edges that spectrally approximates G , in the sense of Spielmen and Teng [54], 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 [53] and uses results from recent work on space-bounded Laplacian solvers [43]. 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.