Spectral Sparsification via Bounded-Independence Sampling
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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 Õ(klog (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/ϵ^2) edges that spectrally approximates G, in the sense of Spielmen and Teng [ST04], 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 [SS08] and uses results from recent work on space-bounded Laplacian solvers [MRSV17]. 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.
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