2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS) (2014)
Philadelphia, PA, USA
Oct. 18, 2014 to Oct. 21, 2014
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2014.66
We present the first single pass algorithm for computing spectral sparsifiers of graphs in the dynamic semi-streaming model. Given a single pass over a stream containing insertions and deletions of edges to a graph, G, our algorithm maintains a randomized linear sketch of the incidence matrix into dimension O((1/&epsi;2) n polylog(n)). Using this sketch, the algorithm can output a (1 +/- &epsi;) spectral sparsifier for G with high probability. While O((1/&epsi;2) n polylog(n)) space algorithms are known for computing cut sparsifiers in dynamic streams [AGM12b, GKP12] and spectral sparsifiers in insertion-only streams [KL11], prior to our work, the best known single pass algorithm for maintaining spectral sparsifiers in dynamic streams required sketches of dimension &#x03A9;((1/&epsi;2) n(5/3)) [AGM14]. To achieve our result, we show that, using a coarse sparsifier of G and a linear sketch of G's incidence matrix, it is possible to sample edges by effective resistance, obtaining a spectral sparsifier of arbitrary precision. Sampling from the sketch requires a novel application of ell2/ell2 sparse recovery, a natural extension of the ell0 methods used for cut sparsifiers in [AGM12b]. Recent work of [MP12] on row sampling for matrix approximation gives a recursive approach for obtaining the required coarse sparsifiers. Under certain restrictions, our approach also extends to the problem of maintaining a spectral approximation for a general matrix AT A given a stream of updates to rows in A.
Approximation methods, Heuristic algorithms, Approximation algorithms, Sparse matrices, Laplace equations, Computational modeling, Resistance
M. Kapralov, Y. T. Lee, C. Musco, C. Musco and A. Sidford, "Single Pass Spectral Sparsification in Dynamic Streams," 2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), Philadelphia, PA, USA, 2014, pp. 561-570.