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 ɛ > 0, our algorithm runs in space \(\widetilde{O}(k\log (N\cdot w_{\mathrm{max}}/w_{\mathrm{min}})),\) where w max and w min are the maximum and minimum edge weights in G , and produces a weighted graph H with \(\widetilde{O}(n^{1+2/k}/\varepsilon ^2)\) edges that spectrally approximates G , in the sense of Spielman and Teng, up to an error of ɛ. 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.